Il Principio Di Minimo E Sue Applicazioni Alle Equazioni Funzionali (C.I.M.E. Summer Schools, 17)

.

,

<

<

where G i8 indepettdeut of u . Tlie lemma is easily proved; for R 0 it is merely the usual ii~terpolati011 iueqnality for L, norms. Let us tarn now to the proof of the theore~n, or a t lea.st to tile main poir~ts. Consider first the Sobolev ease, a = 1. It saffices to coiisider the case j = 0 , 111 = 1, frorri mhich the ge~teralresult ntay then be derived. If r n (2.2) asserts that u satisfies :I oertaiu Bolder coi~dition,A I I ~a11 elementary proof due to Morrey has loug been k~toan.We shitll sketch it here for functiorls defined ill a genet,al dorl~airi (i3. De.fittition : A domaill (2) is said to Itave the s t r o ~ gcoue property if there exist positive constauts I / , i;r~ida closed solid right sghel.ica1 colle V of fixed opeuing aud height sue11 that ally points P , Q i n (the closure of q)with

>

>

a

are vertices of coues Vp, VQ lying in (i3 which tire courguent to V and have the followi~~g property : the volume of the intersection of the sets: Vp, VQ, alld the two spheres with contor8 P , Q w,nd radius P- Q 1 , is not less than R P - Q We 11ow prove the assertion If u hne first tlerivntiver in L,., v > a, is n dowtailc Q hanoing the strong cone property, then for points P,Q i ~ r9 with L' - Q 5 a we h.ave

I

I

In.

,

I

I

,

wltere the constant depends only on a , 1 , V n nird (From this follows easily an estimate for [a]

I-;

.,

1..

depending on the

domain). Proof: Set 8 = P - Q and let Sp(SQ) be the intersection of Vp ( VQ) with the sphere about, P ( Q ) radius 8 . Set BP n .SQ= S If R is a poi~lt in IS we have, OII i~ltegratingwith respectt to R over S,

I

I

.

Volume of 8 . I z c ( P ) - w ( Q ) I 5

+I/

I

Izc(P)-a(R)ltlR+

S

- u (Q) l d B -

r (4

S

Because of the strong cone property the left l~andside is not less than

The first term on the right may be estilrlated as follows. Introducing polar coordinates Q , 7 , about P , where 9 is A unit vector, we fiud easily that the first term in the right is bouuded by

(where d o is the element of area on the unit sphere, and d o is the element of volume)

L. NIRENBKRG : 012 elliptic partial

by Holder7s inequality,

/ 1 lvd

< constant . s

? (SP

-

I

A silnilar estilnate holds for the t e r ~ u

I

$)+*

a (R) - 16

(0) Id

12

, and

the

. ' i

result follows. We retrlrr~i ~ o mto f i ~ a c t i o ~defined ~s in the fill1 11-space. Soppose r vc. We shall prove a strouger for~nulatioll of (2.2), namely

<

r s-1 <---nlel?;. 2 n-r i 8 5 ,

IU,,,. 11-r

< <

,

For 1 r vc (2.4) follows from the special case r = 1 as o11e readily verifies, by sir~lply agplyil~g the illequality for v = 1 to the fuuction n-1 -

V = l u le-rr aald u~illgHiilderls illequality in a sniixable way. T h i ~ sit silffices to prove (2.4) for the ease r = 1

.

We shall prove (2.4)' here for

where

I,

28

= 3 . One sees easily that

denotes illtegratiou i110~gtile ftdl lil~e tllroiigl~ x parallel to

i

the xi axis. Thus

Integrating with respect to xi the11 x z , and the11 x3 me find with the aid of Scl~marz~s inequalitmy

and finally

that is, (2.4)'. For general n the inequality i~ provetl in the same way wit11 the aid of Hiilder's ineqru~lity. Suppose fiually, for j = 0 , r t = 1, tllat = 98 ; tllis is the exceptional case 2. We claim that

.

where the const,ant tlepends o~llyn , q i~nd p I t sllffices to show this for large p and tltis is e;~sily done by rrppliyng (2.4)' to the fulrctioli v = 1 u l p ( l - l i a ) , and using Hiildel.'s illeqltality i t 1 a jutlic'l ~ o ~1n11 i s tl~ler. ~ e ' us t now consider the otlter extreme case a = j / m . I t suffices to cousider the cAse j = 1 w = 2 , the general case mtly then be proved by iuduction on m . We clrrinl tlli~tthe followiug lrolds

,

1

(2.5)

1

I D u l < e l Z l 2 u l 7 / u l : for P-

2 1 1 -=-+-; p 9 . q

l < q , v ~ m , -

with c nth absolute cottxtant. Incide~ltally, :IS. Utlgar poillted ont, we ala,y pennit q to be any positive nurnl)er, 1)ut I sllall cotrtitle rnyself to the case cited, in fact to the case q finite, 1 r 00 The gelleral case may be obtsiued by a slightly difft$re~lti ~ r g ~ ~ ~ t lor e l ljnst t , by lethitrg q tel~dto a, and 1. teud to 1 or m ill (2.6). Inequality (2.5) follows fro111 the corrisl~ot~dinginequality in one dimension

< < .

which ltolds for the full, or Italf-infinite line (with c an absolute constant), by iutegrating with respect to the other vt~riablesand appliyng Hiilder's inequality. Our proof of (?.6), though elementary, is slightly tricky. Peter Ungar has found another slightly longer proof wliich furnishes a better value for c. The proof is bssecl on }L sinrpla lemma wllicll we leave as a,n exercise.

L. NIRENB~RG : On elliptie partial LEMMA: Oa an intevval 1, whose lenglbt we also derrote by 1, we have

wit8

o

an absolute co~tstnnt. We s b l l prove tlrat for any interval L : 0 < x inequality holds

L the following

(2.6) follows easily from (2.8). I n proving (2.8) we may suppose that 1 ttx, ,1 = I . We shall cover the interval L by a finite nurnber of successive intervals A,, 12, each one having as initial poilrt the ead poiirt of tlre preceditrg. For k a fixed yosi-

... ,

L k the right of (2.7) is greaber thiun the

tive integer, clroose first the i~rtervel1:0 (XI-, aild consider (2.7) for this ieterval. If the first term OII second set 1, = 1; we then have

since I urn ,1 = 1. If however tlre seco~rdt e r a of (2.7) is the greater extend the interval 1 (keeping its left endpoint fixed) 1111til t l ~ etwo terms of the right of (2.7) becotne equal. Since 1

+ 8, - Er >

0 eqnillity of these two

terms nils st occnr for a finite value of 1. Let 1, be the resulting interval. We thei hive

Starting a t the end poii~t of 1, repeat this process, keeping k fixed, choosing A,, 1,, until L is covered. There are cletbrly a t most lc such

...,

iutervals

4.If we now

r

sum our estimates for

J hi

IP d r

we fitrd, with

differential equations

-+ 2r

the aitl of Holder's illequality (recall that P

If we uow let to zero, bec:ruse

16

-

w tlte first term

011

> 1 , at111we obtain (2.8),

P = 1) that 2p

the right of the preceding tends completi~rgthe proof of (2.6).

Lecture 111. The Dirichlet Problem. We co~~sider now elliptic tlifferential operators, confi~ringourselves for simplicit,y to a single equw,tiolr for one unkr~own. Let L ( z , D) be a partial differel~tiill ol~erator with co~tlplex valued coefficients, w.nd let L' be the part of highest order. L is elliptic if there are no real clraracteristics, i. e;, L'(x,fl+O,

real E + O .

I t is ettsily see11 tlrat: for more tlrat~two vari~tbles,a > 2 , ellipticity implies t11at tire order k of L is evetl. In treaitiug tile Dirichlet problem we shall assume that k = 2 918 is eve]) and hirat the operi~toris strongly elliptic, i. e. that (efter possibly mu1tiplying by a stlitable coinplex function)

The Diriolrlet problem

consist,^

of finding B solution in a domain

9of

where a/& represeuts differentiation l~orlnalto the boundary. Here f and Q,, are given functions in 9and 4 respectively. We shall describe here the Hilbert space approach to the Dirichlet problem, which is based on sotne form of the projectioll theorenl, and is related to the classical method of minimizing the Dirichlet integral, In its

L. N I ~ ~ N B ~ :C On R Gelliptic partiat 0

preseut form the existellee theory is ~naiulydue to Garding, Vislrik, Browdela and others ; we refer t l ~ ereirder to [9] aud [a]for expositions and refere~rces. This and the followi~~g lecture comprise a brief description of [9]. The 0 theory is based on a sitrgle L2 inequality. Gardi~rg's inequality, expressi~~g the positive defiuiteness of the Dirichlet integral associated with the differential operator. Since this approach to the Dirichlet problem requires cousiderable differentiability assumptions on the coefficients we shall ~ssilmefor simplicity that they are of class Om in cZ) and that the boundary d is sufficiently smooth. We shall also assume Q to be bounded. Parthermore if the Qj are sufficiently smooth we may subtract from u a furlction having the same Dirichlet data as u, so we shall consider the case where the Qj vanish (3.3)

Lu=f

The Hilbert space approach yields at first
wlrich belongs to C r (Q), i. e. is of the class Cm and has for every compact support ill 0. Here ( ) denotes the L, scalar product, and LX is the formal adjoint of L . I n addition to the L, norm we also introduce 0 the Hilbert spaces Hj(Hj), j a non-negative inleger. These are the closures in the norm (using the notation of Lecture I)

,

of the spaces C m ( 9 ) (Q). The associated norm end spaces relative to a a a ou subdom~ina will be denoted by 1 1 H, Hi Clearly H,, L2. We remark that for j i Hj c Hi and the set 11 u 1 1 s ,constant is compact in Hi. Pollowiug Sobolev and Friedrichs we say that a fimction u in 9has strong derivatives in 4 up to order j ia 00)if u belongs to H~(H:) for every compact subdomain a of 9. With the aid of the results of the

>

11. ,

, .

=g=

differential equations

>

preceding lecture we see tllat e fiulction ill Hj is conti~luousif 2 j a . Functions h satisfy the bou~tdary conditions of (3.3) in a geaelxlized sense. We now formulate the GENERALIZED DIRICHLETPROBLEM : #ioen j' in H,, fwd a weak so/ution 0 u in Hm of h2c= f . Using the notation of Lecture 1 we may write the operator L in the form L = 2 Dfl ap,,Dy lei, lulsm

2,

.

0

If u is weak solution in H,,, we lnay then carry out some partial iute. gration in equation (3.4) and write it as

,

B [ u , v] is linear in u antilinear in v and satisfies, by Schwarz~iuequality

We sl~all assulne the strong ellipticity (3.2) to hold uniformly, i. e. for some positive constant c, 8.6

(- 1Irn 2

IBI lul-m

ab,, (8)tflP .>c, 1 8 l Z r n

,

8 real,

for all 3 in (D. Our main result is THEOREM:For ssl~,flcientlylarge the generalized Dirichlet proble?,, for the equation (L )': 14 =f admit8 a unique solutio?z. For the eqztntion Lzc =f we have the Wedholm alternative. The .L, estimate on which t l ~ etheorem is based is G ~ B D I N G ~INEQUALITY S : There exist constants c 0 and C such that

+

>

holds for every pl in C: ( 0 ) . This ail1 be proved in the next lecture. I t is clear from (3.5) that tbe 0 inequality extends also to functions in Hm, and it follows from (3.6) that 0 the only solution in H, of (L 0)u = 0 is = 0 . Let us now prove the theorem. Suppose first that the operator is symmetric, i. e. B [pl , p] is real. and that the coastant 0 in (3.6) vanishes - which we may achieve by consideriug L C in place of L . It follows

+

+

L. NIRICNBERG : On elliptic partiat

,

from (3.5), (3.6) (with C = 0) that B [u v] serves as an alternative scalar product in the Hilbert space $,; the norms B [u u] and 1 u 1 , are eqaivaleut. We see that the antilinear functional ( f , v ) defined for all y in in, satisfies

,

and is therefore a bounded functional. By the well known represel~tation 0 theorem there exists therefore a function u in the Hilbert space H,, such that

i~ then the solution of the Dirichlet problem, and we have proved the Brat part of the theorem with C! = C . To prove the secoud part we write the equation Lu =f in the form (L C) = Cu f or

'U

+

+

Since ( L C)-1 maps H, boundedly into k,, it is completely coutinuous in H,, by a previous remark, aud from the Riesz theory for completely continuous operators we derive the second part of the theorem. Suppose now that B [ y y] is not symmetric. If we add C ( y y ) to B so that it satisfies

,

+

,

,

then we may still rely on a generalized rep~~esentation theorem due to Lax alld Milgram. We conclude the lecture with this REPRESENTATION THEOREM : Let B ( x y) be tc jovm de$ned l o r pairs of vector x 7 y in n Hilbert space H(uorn111 which is littear in a , antilinear in y and satisjies

,

,

I/),

Suppose that for some positive cottstant c the inequality (3.8)

I

B (x 7 $1 l 2 c 11 x 112

holds for every x i s H. Then every bounded antilinear functional P ( x ) admits tile representation F ( x ) = B ( v , x) = H ( x , w ) For $xed e1ei)tents 11, ,w to11iclr art: u~iiqne.

.

diferential equations Proof: For any Bxed element v , B (v, x) is a bout~dedantilinear functional of x and therefore admits the representation

,

,

for some element y where ( )H denotes the scalar prodect in H. This defines a ~ n a p p i ~y~= g A v which is clearly liuear. L e t t i ~ ~xg = v and applying (3.8)we find that

It follows that the operator A has a bounded inverse and t l ~ a tits range is closed. Fort,hermore the v corresponding to any y is unique. To see that the rarge of A is the whole space H suppose that z is orthogo~~sl to it. Then we hr~ve B (v z)= 0 for a11 u . From (3.8) it follows, by setting v = z , that z = 0 . Thus A maps onto the elltire space, and therefore every a~ltili~learfunctional ' P ( x ) being of the form (y, x ) admits ~ the representation P(x)= B (V $2.). The other representation is proved in a similar way.

,

,

Lecture IV. A Priori Estimates. Before provil~gGardiugls inequality let us make some general remarks about a priori estimates. Consider a differential equation h =f of order k and assume that the solution has been made unique by some auxilirry conditions. One wauts to study the inverse operator - to see, for instance, to what class of functions the solution belongs, i f f belonge to rt give11 class. For this problem, and also for the existence theory, a priori ineq~u~lities play a basic role. Let us suppose that the auxiliary couditions are homogeneous, then a typical a priori estimate would assert that for some uorm 11 11

11 Dfl u 11 zz

constant 11 Lzc

11

For instance, if we kr~owthat the equatiol~has a solution of class OK for all f uf class 0 then indeed, by a simple application of the closed graph theorem, we would have

II Dfi 21 1) < constant )ILzc 11 ,

1 B I s; K -j

,

L. NIEENBEBG : 0s elliptic partial with 11 11 the usual uorm.in Cj. 111 general if Lu has fit~ite (1 11 norm we will not obtaili such an inequality for K = k , rather K < k ; that is we c a ~ ~ ~ esti~riate lot iudividu;~lly all derivatives enterir~g ill L . However I believe that elliptic equatiolis oil11 be cltari~cterizedas tllose for which oue can e s t i ~ ~ t a all t e derivatives, i. e.

for a wide class of n o r m (this is skated its it co~iviction~ i o ti~ tlieorem). Cousider ttow an elliptic equatiott Lu =f with saitwble lto~nogel~eoas bouriditry coaditions. Most i h priori estimates are just of tlie type (4.1) or, if oue does not assume uniql~el~ess, of the for111

Iudeed ~ n u c hof the theory of elliptic e q ~ ~ a t i ois~ concerned ~s witlt proving s u c l ~estilnt~tesfor various liorms 11 11 a11(1proviug attalogo~lsestimates for fuuctious with no boundary restrietioos :

,

Here a is ally compact srtbdomai~tof 9, and the norm 11 1 " is cousidered only for fu~rctio~is defined in a I A word qf oautiow: The estilrlate do not hold for the most obious norm tliat orhe would try, ~iamelytlie 111axi1n111n (or Go) norlrl lior i l l getteral for Cj nonns, however they do hold for Cj+"iorn~s, 0 a l , and for many illtegral norms. W e quote some immediate colisequence of (4.2), (4.2)'. 1. If f aud the coeffieiet~ts of L are in Cm t l ~ e na solutio~t of Lu =f is also in COO. This follows fairly easily from (4.2)'. 2. 80lutiotis of Lu = 0 with bounded llorlrr 11 11 form a compact family. This follows from (4.2)' itutl the Calculus Levma: The set 11 # (1 11 Du 11 coltstant is conlpuct in the 11 as norm. q a o e with (1 This lemma holds for a wide class of norms. 3. The set of sol~itionsof L.u = 0 sitisfyiug the bouudary conditions (so that (4.2) holds) is fiuite dimensional. This follows with the aid of the Oalculus Lemma. I would like to describe briefly a general recipe for proving such estimaten. This cousists of several steps : 1. I11 case of (4.4)' prove it fil~stfor equations with constant coeffioiente and only highest order terms, and for functions of compact support.

.

< <

+

diferential equations 111 case (4.2), prove it also for such eqnations and for functions defioed in a half spitce, v;cnishing ueir infi~lity,and satisfiyng (on the p1au:lr bouudary) the boundary conditions. These are also assrimed to have c o ~ ~ s t a ncoeffit cients (i. e. to be transliltio~~ invibriant). 2. Now elitnit~atethe hypothesis of compact support. 3. Extellti the estirn~teto variable coefficients as follows : with the aid of a partitions of unity write the funchion u as a sum of fiinctions % with s~nallsupport, in each of which the leadiug coefficients are close to coastiints, ant1 treat the variation from aoustant as an error term, using the results of Step 2 end the followiug lemma which lnay also be used ill the proof of Step 2. Caloz~lusLumnza : For appropriute constants o, o,

,

where for ,/unctiotu of compact support we ?nay take c, = 0 a d c, indepejtdetzt o f the support of u This holds for a wide class of norms. In case the support of ui touclreu the boundary, make a local change of variable to flatten out the bol~ndaryso t h ~ ,Steps t 1 ; ~ n d2 can be i~pplied. Tlre main step h'ere is Step 1. We retnark that in Step 3 we rely on ( t i t , least) the c o ~ ~ t i n ~ i iof t y tlre leading coefficients of L or on the fact that tl~eydiffer little from constunts in small domains. Because of this olle does not obtain in this way the Inore refined estimates required for tt.eating nouliue~r probletns, such as those in Bers, Nireuberg [lo], de Giorgi [li], or Nash [12]. T l ~ ellor~rls for which such estimates are easiest to derive are the L, norms for functions and their derivatives, and we shall illustrate the recipe 0 for these by proving Garding's inequality in its general form. Consider a quadratic integral form d e f i ~ ~ efor d Om functions with compact support in a bouuded domain cZ)

.

,

and suppose that the (complex valued) coefficients c b , are cotrtiuuol~sin (i3. A necessary and sufloient condition jor the existence of positive constants c ,0 so that the ifiequality

L. NIKINBEKG : On elliptic partial

holds for all u E C r (9i) s that for some positive cotbsttrtlt co 3e

2

IS1 > lyl-ns

O P , ~5P 57 2 c0

I 5 12m

for a11 real

5.

Here the notatio~lsof Lecture 3 is used. I'roof: We prove first the sufficiency, following our recipe. The Calculus Lelr~rne(4.3) will be used in the forlrr: For every E > 0 there is :I aol~stant C(E)such that for every Gm function u with compact support

Tl~isis contained in our inequalities of Lecture 2, but is most easily proved with the aid of Fourier transforms. We consider now t l ~ edifferent steps in provir~g(4.5), the Step 2 of the recipe doe8 not occur here since our fnrlctio~~s have comlract support. 1. Suppose that the oP,, are c o ~ ~ s t t tat111 ~ l t vr~irishunless IbI = l y l = m . We ilrtroduce the Fourier t,ral~sformof u

By Parseval's theorem we have

proving (4.5) for this special case. We uow consider' the verii~blecoefficie~rtcase and breitk Step 3 into two parts. 2. Suppose that the support of u is sufficieatly small, contained, say, in a small sphere about the o r i g i ~ .Then accorfing to the preceding inequality we have

diff6rsntial equations

If now the support of u is so srr~allthat cp,, hi18 smi~lloscillation there we see that the second term on tlre riglrt may be bounded by

The third term is trivially bolrnded by oo~lsta~it 11 u ( I , Thus we find that

(1 u

from which follows the inequality

(4.5) now follows with the aid of (4.7). 3. Consider )low the general case. Uonstruct e partition of unity in

with the support of each wj as small as desired. Then

by the preceding Case 2, 2 constant (1 u 1 ;

+ 0 (11

u I)m

. )Iu (In,-1)

and the desired result now follows easily with the aid of (4.7).

a,

L. NIRENBERG ': On, elliptic partial

,

We see that the constants c 0 iu (4.5) depend OII c,, a11 upper of the leadi~rg bonr~d for the cg,, 1 , i111d 011 the modulus of co~~tiunity = y I = n b , a11d finally on tlle iion~ail~ 0 . cBly with I Now for the proof of the nacessity of (4.6). Suppose that (4.5) holds aud that the left Itand side of (4.5) va~~islles for solne real I[ I , and some poiut ill (i3, say the origin. Followiug the ergument iu Step 2 in the proof of sufficiency we see that tlie inequl~lity

I I

I

[=r, I/=

llolds for all Cw u with support in sonle fixed neighborhood U about the origin and in (2). Set u = e""ra. c (a) for real I where 5 (x) is a fixed real Cm f u a c t i o ~with ~ s~rpportin U and in 9 , One sees reitdily that as I w the left hand side of (4.5)' is 0 ( P I ) and not o (IZm)while the right hand side is 0 (1%"'-I), so that (4.5)' does not hold. C Garding's inequality (4.5) is a t one end of a whole spectrim of interesting and useful ineql~alities making tliffere~ltrequirelr~entson u at the 0 bou~ldary,Gardiugls inequality niaking the maximill reqaire~nent- that a t the boundary. A t the all derivatives of u of order less than nt va~~islr other end of the spectrunl is the ir~equalityof Arouszi~ju[13] involving no boulrdary conditions whatsoever. Aro~lszaju co~lsiders a 11ulnber of dillerential operators Lj (8,D), j = 1 , N of order ~ 8 with , c:oefficients eoutinuons ill the closure of a bou~~detldomain (a, and solves the following problern : Under what conditions can one assert that for 1111 C" functio~lsu in 9 the inequality

,

-

... ,

holds, with ttle ao~lstant iudepeudeut of , u ? He gives necessary and sufficieut conditions : (a) the operibtor 2 Lj L; is elliptic, here Lf is the formal adjoint of L j . (b) A t any boundary point - 8 of 9 , if is the unit normal to ci) m d 5 . 10 is any real vector taugent to 6 then the polynomials in z , -+ Lj (x, 4 z n) have no common complex root z . Here Lj is the leadiug part of Lj An exs~nple of Aronzsaju7s illequality is the following; for functions u (x, y) in a bounded domain in the plane

+

.

J1

uW 12 ilx dy

sz constant

I

(1 urn l2 + 1 u, l2

+ I u le) dx dy .

diffevential equations Even this simple extcmple is not trivial to prove. Since the report of Aronszajn a 11nmber of people Ileve coottsidered the problem *of proving (4.5) for v n r i o ~ ~quatlrat,ia s forms (4.4) and under various differential boundary oo~lditions. For one operator L j Agmon, Douglis, Nireuberg [14], (in a fortheotning paper which will be discussed later) have characterized these dilTerential boundary coilditions whicb are m/2 in unmber aud for which (4.8) holds. Scheclrter [l5] has treated N operators a l ~ dgeneral boundary conclitions. Aronszaju, in i ~ n p ~ ~ b l i swork, l~ed has treated the geaeral problrtn (4.5). Also Horl~landerand Agmon [l6] have solved the general problem for (4.5) a,nd genelnal ciifferel~tiiclboundary conditions. The proofs follow the recipe outlitled above, the trraii~step being the first, for functions in a llalf spibce. We conclude the lect~ire with a result that will be used in proving the differe~rtiabilit~y r ~ t the boundary of solutiolls of elliptic equi~tions. Iu the followil~g ,ZR denotes the henlisphere I x I R x,, 2 0 We shall denote the varia,ble x,, by t (xi, x,,-,) by x and (xi ... x,) by (x, t) . Lemma: Let u be a weak solutio~tof a diflr.etttia1 equatios (of order k) with, for simnplioity, CW coeflcienfs,

,

< ,

... ,

, ,

.

i n the interior of a hemisphere Z R , where .b are giver&f~cnctions,mid assultre that the plane t = 0 i s ~iowhevecharacteristic, ill jirct that the coeflcient a of I$ i n L does not vtrlzish. I f fol* every 6 0 the funelions .fP,IIb lc fbr I /?1 <j , D, Dj w belong to Lz i r ~2R-sthen trlro the ftrt~ctioftDft' 11 hcts this property. For j 2 k - I there is notl~ing to prove, as we may solve for the functioll ~ ( " z c froin the differential equatiou (4.9) operated on by D::+'-~. Thus we suppose j k -1 . The proof ltlakes use of a well known formula giving explicitly a smooth extension of a function v defined ill a half spitce t > O to a fu~~ction defined in the full space:

>

<

with the ;lj chosen so that

L. NIKENBLEHG : On elliptic partial

28

We observe that,

Here the norm on the left is over the full space while on the right it is over the half space t 0 A o o f of the Leircma : Choose a fixed 6 0 , let [ (s t ) be a fixed Cm fuaction with sapport in I x l2 t2 R2 and wirich equals one in , Z R 4 and set [ au = v . If we can prove that D/+' v belongs to L2 then, since a+0 it follows easily that D/+'u is ill L2 ill 2R-S. FPOII~ our assumptions we see that v is a weak solution of a differential oquation of the form

> .

+ <

>

,

,

where the us,, b e l o ~ ~ so g L,, autl that derivatives D, 1)jv and v itself belorrg to L, ., For N sufficierltly large we now exterrd tire filrrctiolls v , v,,, to negative t defining UN by (4.10)a l ~ dv , , , , ~by

,

One may then verify that the equation

holds in the entire 8pace iu the weak sense, R I I ~that tire v , , , , ~ the , derivatives D, DjvN and V N itself belong to L 2 . Let us now take Folirier tr:rr~sforr~~s with respect to 8 and t , and write ( E l , E,,-,) = 5, En = t Denotirrg the trsasfonn of a function f by we find that

...,

7

w

.

GN a r ~ d1 5 1 (1 E I j f I t I j )

,

belougirrg to L, in the (E t ) spice. To conclnde the proof we have to show that belougs to L 2 . To this end write

with

VS,,,~,

differential equations We shall show tJhat ertch term on the right belongs to L,. Pron~(431) we find that the first tern1 on the right is bout~tled by

Since s $ I y I < k -j - 1 it follows t h ~ the t fact,or of v , , ,is~ uniformly bounded, and hence t h i ~ tthis term belongs to I,,, since the V,,~,N do. The second term on the right of (4.12) is bounded by

with o an absolafe constant, and hence belongs also to L , , by an earlier remark. This completes the proof of the Le~n~na.

Lecture V. The 1)ilferentiabilitj- of Weak Solutions of Elliptic Equations

In this and the next lecture we slrall preseut a self contained proof of the well known result thiht solutions of elliptic eqllations with C m coefficients are of class Cm. Many proofs exist in the literatoi~eincluding proofs for Inore general 1171, Malgrange [18]. The proof here classes of equations, see Hiirn~;u~~ler seems rather straigtforward; it i~ based essent.ially OII a proof given by Lax [19] and is closely related to proofs give11 in lectures by Bers [20] a ~ l dSchwartz [21] (see also [9]). We confine oiirselves as before t,o a single equation (not l~ecessarily strongly elliptic) although the argument extends also to systems. Dieretitiability !L'heorenz : If u is a locally epucrre ittteq~ableweak sohtion of the elliptic epuatiofo L u =f, alzd f E Cm tlbelz u E Om. Remark : If u is a distribution solution theu u = Ak v for some continuous v (here A is the Laplace operator), and v is theu a weak solution of L A h =f . The Theorem holds therefore for this case also. The proof consists in showing that u has L, derivatives of all orders in every compact subdomain. That u E GM" tl~en follows from the Sobolev estimates proved in Lecture 2. However since we ollly need a very simple case of the Sobolev lemmas me present a separate proof of it here.

Lemzcin (Sobolev): In n u ~vnoothu dot~aaitt9 i f u hrrs I;, derivtctives up to order s in for s 9812, thes u is continttous i n 0 . In fact

a

>

Proqf: The first assertion follows easily from tile iaequality. To prove the i~lequalily let x, be an inner point in 0 (for 8irlrplicif.y take x, = 0) alld suppose tlrere is 'a sphere about x,, in 0 with radius R . Let furthermorl 5 ( r ) be a function iir Cm, e q ~ l ~tol 1 for 0 T L R I 2 , and vanishing for r 2 R By ii~tegratiou alolrg ally rl~tliusfrom zo= 0 , aud by repeated partial integration we see that

<

.

iutegrating over the unit sphere (with area Q) of radial directions one finds

>

using ScLwarz i~leqnality.For. s n/2 the last iutegral is finite. If the boundary of 0 is such that a t any point in there exists a cone with a fixed ope~iiogalrd length contailled in then tire same proof holds; instead of illtegrating over the full sphere of radial directions, we merely integrate over the directions lying in the cone. The proof of the Differentiabilih Theorem co~lsistsmainly of a series of simple lemmas of c a l c u l ~ ~coucer~~ed s with a special situatioa) that of periodic functious, and tlris lecture will confirled to these oalculus statements. We consider peviodic fulrctio~rsu E C* with period 2 n in each z,. For such f u ~ ~ c t i o utire s Fourier series ,u=XuEei..f, 5

( f j = in tegor) converges uuifornll y.

E = (E, . ... .5,,)

By Parseval's equality me have the following estimate for each nonnegative 8 (5.1) ,

+1 E

canstant 2 (1 5

+ I iI?

I - constant 2 (1 C

)S( U E I'

where the i ~ ~ t e g r aisl taken over :L period cube. For any integer 8 we introdl~cethe followiag scalar prodnct and norm, differing slightsly from our previous notation,

,

,

We write (u u), = (u u ) and proceed with the Calculus : 1. 11 u Il is inocretisillg in 8 Flirlhern~ore for t, E 0 there is a constant C (6) such that

.

>

< < t, 8

and any

< +

Proof: For any a 2 0 , as E ot9 C ( 8 ) ot1. 2. Set p , = ( l - A ) t u , y = ( l - A ) t v , sethatp,=2ue(L+I&12)teb.~. e From this we bud

As a consequence we have Lemma : If o E Cm, then (5.5)

Proof: find

,v)t = (a W v)t 4- 0 (I1 llt I1 v + I/ Assunie t < 0 . Using (5.4), (5.3), (5.1), and (W 11

1

91

]It-1

Ilt-I

partial integration, we

( w u , ~ ) ~ = ((1 w- A ) - t p , ! ~ ) = ( ( l - A)-tp,

Iu the case t / 0 the proof is similar.

I1 v (It).

zy)

L. NIRENBERG : On elliptic partial 3. Sckwar(t)t'a inequality : (Clear I)

Proof: According to (5.6) tile left side of (5.7) is 11ot smaller than the right side. If however we set v = (1- A ) t t z , then, by (5.4)

proving (5.7). We can now form Hilbert espace Hs by colnyletil~g Cm fullotions in 11,. For a > 0 these agree wit11 our previous definitious. the norms Obviously H,c Ht f6r a t All the previous results llold for fuuctions with the appropriate i~orlnsfinite, for insti~~lce (5.7). We lnay regard H, as give11 by a forlnal Fourier series with fiuite 11 11, noun. We remark that the scalar product

11

>

.

,

,

is defined, by extension, for any fiu~ctions11 E Hg v E H-, and that ally bounded linear fu~~ctional f (u)defined on Hs lnay be represented in the form

with v E H-,; this follows imlnediately from the Fourier series representation, so that we ]nay regclrd H-, as dual to H s . Though me shell not use this, we reulark thirt the closed ulrit ball 11 u 11, 1 in H, is compact, i l l Ht for s t We continue with the calculus. 4. Consider any differential operator L of order k with Cm coefficients.

<

>.

Claim :

More precisely

, ,

where c = G (k u) K is a bound for the leading coefficieuts, and K' is a bound for all coefficients and their derivatives u p to order 1s 1.

P ~ . o o f :S i ~ ~ cobviously e 11 Di 24,II 5 C O I I S ~11 14 prove (5.9), to show that i f a E Ca theu

1

it soffices, in order to

I

where k' a11t1k" are bounds for a 1 aud 1 Dj (1 (j 1 s 1 ) respectively. P ~ o o fof (5.10) : Co~~siderfirst the cave s 0 . Set 97 = (1- A)' ly' = (1 A ) S a11 theu we I~iive,by (5.4), and partial integratiou,

-

lntegratiug t l ~ el ~ , s by t pt~rts(-

So dividiug by

11 y

8)

<

t~

times we tind it is not greater than

we have, with the aid of (5.3),

=o k

11 [ I s 21

+ c k' 11

16

Ils-l

by (5.3).

I u case s 2 0 we have

IIauII:=(a u , (1. - A)'au) and may integrate by parts as above. So L car1 be extendetl to all of Hs and maps it boundedly into Hs-k. This operation of L agrees with that of L actiug on u , regarded as a distribution. Teckdcal Lemma: Suppose w is a Ca real function, then

To conclude this leoture we consider Diferenoe Quotients: For given vector h let

L. NIRIENBIERG : Om elliptic partial be the difference quotielit. One verifies easily : 11 u (8

,

+ h) 11, = 11 u (x)111,,

<

.

Furthermore : If tc 6 Ha uh E H, and 11 uh 1 , k for each h, then )I u llsfl 5 k Corolltrry :If t c E H,,I1 u: 11 k for each 8 then a E H,+, and 11 u I ,+, 5 k. Proof; Let u = 8 t c c &.a7 and let u~ = 8 uc e b . 5 . One filids

<

,

IblSi

€

Lecture ?I. Proof of the Differelltiability Theorelti. Let now L be an elliptic operator of order k . I n the pe~,iodiccase we prove the Differelltiability Theorem in the form D~FFERENTIAB~L~TY T I I B ~ ~ ~ E Ij' M :11 E H, L u E H s - k + ~ tlteu IL E Ha+, So it. follows that iJ",rE H, atid L u E Ht-k, thelt. t i E Ht The non-periodic case is easily reduced to this as follows; We prove successively that u has L2 first order derivatives, thee second order derivatives, then second order derivatives, and so on. To cwry out this reduction let [ be a C m fnnetiou defined ill a neighborhood of s poiut and with colnpact support. Let v = [ 14 and extend v and the coefficients of L s s periodic functions. SO

,

.

,

.

where f = [ L t i , g = L (( u)- ( L TL ; g contai~~s only derivatives of t c up to order k - 1 , and so, as is easily seen with aid of (5.8) llas finite 11 111-k norm. So L v E H l-k, therefore u E Hi and so u has L, derivatives in a neighborhood of the point. Using this one repeats the argn~nentfor a sn~aller so v E H 2 , itnd so on. neigborl~ood,and sees that 5 v E The proof of tlre Differentiability Theoreni i l l tlie periodic c u e follows 0 easily, in turn, from the followi~igestin~iitewhich is tlie analogue of Garding's inequality. Basic E~timata: For any s a,,

,

,

11 u lls+k < constant 11 L u [Is + constant 11 18 ,1

.

Postponing the proof of (6.1) let us prove the Differentiability Theorem. Consider a difference quotient uh. If Lh represents the operator obtained by replacing each coefficient in L by its differeuce quotieut we see that h h =

diferenlial equations

-

= ( L w ) ~ Lh.u (8

11 uh ] I s

+ h) Thus we have, from (6.1)


11 Jls-k f constant, 11 u 11, < constant 11 1,u lIs-k+l C constant ( L 2 -

~

)

~

+ constant 11 Lh u (x + 1 ~ ) + co~istant11 u [Is

by (5.1 2): (5.8)

The desired result follows from the corollary after (6.12). Proof of the Rnsio Estimate : Clearly only so s $ k is of interest. The proof cotrsists of several steps, following our recipe, ant1 the proof of G~~rdiug's inequality in Lecture 4. 1. L has constant coefficients with leading terms only. Then

<

0

I l 2 1 l jZk

> constant 2 ua I

while Hence

11 L u 1 ;

+ 11

tc

)>:I

constant 2 ( I

= constant 11 u

(1

+ I l JZ)s

l 2 ( 1 + I 612)S+k = .

2 1 ~

2. L has variable coeffieier~tswit11 leading coefficients differing from constant value by less than 8 , E sufficiently small : Let L, be the operi~tor with these constant coefficients. By case 1. we have

11 u I l s + k I coustant 11 Lo u 11, + aonataut 11 21 I, < constant (11 L 21 1 , + 11 (Lo- Lu) 118) + constant 11 21 1 , < constant 11 Tr u 11% + constant E 11 u Ils+k + constant 11 21 Ils+k-l, by (5.9), 1 (constant 11 L u (1, f co~iatrute 11 u ll.+k + 11 u + constent 11 tc 1 , by (5.2), from which (6.1) follows. Note: If tc has its support in a small set then the leading coefficients, being continuous, difrer little from constant values. So (6.1) holds iu that case.

3. General case: Intrnduce a partition of unity over the closed period cube

with each w , having its suppokt in a small region.

< constal~t2 (L wj u, L a j us)+ constant 2 (1 @j u i j

+ 0 11

( u l18+k:

11 u ((s+k-~)

(since wj u has its support in a small set)

<

- constant

+

3 (1 5 21 112 1 (I

+ II 0

- , by

(5.2)

from which (6.1) follows. We co~~clude this lecture with some remarks concerning the differentiability uear the boundary of the solution of the Dirichlet problem obtained in Lecture 3. Using the notation of that lecture we recall that the solutiol~of L u = f belonged to the space H,. Since the divcussion is local we may assume that the bou~ldaryis given by zN= 0 and thitt we have a solution of the equation in the hemisphere Z R (see Lecture 4). I t suffices, by Sobolev, to show that the sol~ition u has derivatives of all orders in L2 in Z R 4 , for f in Om. We shall merely indicate the first step the proof that u has derivatives of order (st 1) in L2 (see [9]) and sllsll use the notation of the l e ~ n ~ nata the end of Lecture 4. By that lemlna it suffices to prove that derivatives of the form D, Dmu belong to L2 in .ZR4 for any 6 0 . This we do with the aid of difference quotients as above. For fixed 6 0 let 5 (2,t) be a Cm function with support in I x "1 tt< R2, and which equals one in ,ZR-d. Since the function u satisties

+

-

>

>

for all g, ill

H, (ZR) it fo11ows that the function u = 6 u satisfies

for such g,. If we IIOV form difference quotients v h as above with A parallel to the boundary t = 0 we find easily that

.

for cp in iWr (Zx)Setting g, = v h and applying ~ i r d i n ~inequality ~s for strongly elliptic operators of Lecture 3 we obtain a bound for

,

which is independent of h and it follows easily that the derivatives 1lm are in Lz hence that D, Dm u E L2 in ZEl--g.

,

v

Lecture VII. A l'riori Estimates Near the Boundary. 111 the relrii~iuigtime we shall discuss briefly the derivation of estimates Ileiir the boundary for solntio~rs of elliptic eq~~ations iu, for simplicity, it bounded domain (3'. Thihl material is taken from a paper by Agmon, I)onglis, Nirenberg [14] which is coucerlrecl wit11 both Sohauder a ~ l dL, estimates near the boundary for solutions satisfying general boundary coutlition. As remibrked in Lectnre 4 the estimates for p = 2 are special cases of more general results. We wish also to draw attention to a pitper [52] by Honnander concerned with eqi~ationsL u = O with constant coefficients in a lralf space, aud solntious satisfyil~ga number of bot~ndary coliditiolis Kju = O described by differeutial operators Bj with consta~lt coefficierlts. Horma~lder characterizes i~ll'such systems for which the solu. tioris belong to Om on the boundary, and also those for which the solutions are a~lalytica t the boundary. Since the time is li~i~ited we shall restrict ourselves here to the Dirichlet problem for a single elliptic equations of order 2 m

where

represents the uuit normal td the boundary.

L. NILLIENBIHG : On elliptic partial

The operator. L will be required to satisfy a certain condition. Condition O I L L : lf L ' ( x , D) is the leading ptrvt of L , we reqrive that for every pair of independent real vectors ti,t2the yolysoniiccl i n r

huve ezactly m roots oft either side of the real z axis. In three or more di~rlensiol~sthis conditiol~ is automatically satisfied. We see however that the operator

(&.+i &y

in two dinlensions violates

tlle condition. (This operator and others in two di~nellsiollscome under the theory when treated as a system). 111 describing the estimates we make use of t l ~ efollowing norms and selr~iuorms.In lecture 2 we already met the Holder norm, for 0 < a 1

<

For functions of classe Ck in (i3 we also introduce (differing from the notation in Lecture 2)

where the 1. u. b. is taken over all derivdtives of order k , and a11 points in (i3. 111 addition, for functiol~sin C k with Holder oonti~ruous(exponent a) derivatives of order k , we introduce

where the max, is taken over all derivntives of order k . The space of fiumtions with finite I llorlrl is denoted by Ck+" ((i3). (We also uae the notation of Lecture 3 and 4). For functions p, defined on the (smooth) boundary @ of (i3 we also have analogous ~ ~ o r m sdetilled , ill a tather obvious way en terms of local coordinates, and which we deuote in the same way.

Ik+,

diferertial eyuations We now summarize the resnlts without specifying the exact smoothness oor~ditionsqn the boundary; k will denote a non-negative integer. The integral estimates will be stated only for p = 2 . L, Estima.tes : If uE H2, and sntis$es, for simplicity, homoye~ceous Dirichlet dtrta, i. e. y j = 0 , aiid if L u E Hk, aicd tile coef3oients of L belong and to Ck, then u belongs to

Similar reszilts hold jbr equations in integral, or variational, form. Sohauder Estimates :I j y o r some positive a 1 ,uE C 2m+"((i3),LU E 0 k+a(q), yj E C2~$+~+l-j+~, awd the coefjcie~tsof L belong to C ~ + Rthen , u E C2m+Na and

<

Si9)tilar result^ hold for vqrctctions in varicctional form. fiom these oae rcay deriue, tior instarcce, the following result for solutions of L u = 0 under srtitable srtoothqaess &ssustptions on the coafjcieuts : I f u E C ~ - l + ~ ( gand ) y j E Cm-J+k+d,j = 1 m , the% u E Cm-l+k+a @I and

,

,.,. ,

(7.4)

I u Im-I

+k+a 5 constant (2I

Im-j+k+a

+I

10)

*

.

Tlre coustants in the above are indeper~dentof w I n case of uuiqueness of the solntion in the class considered bhe terms (1 u [Io or / u ,1 may be dropped. Mirauda [23], using results of Agmon [25], has recently proved au extended maximum principle for solutions of strougly elliptic equatioas (7.1) in two dimensions, which asserts that (7.4) holds for k = a = 0 I believe that this holds true in general for operators satisfying the condition on L . Tlle Schauder estimates have a number of useful consequences. With their aid oue may prove the existence of solutions of strougly ellilbtic equations having merely Holder colltinuous coefficieats. I n particular, with the aid of (7.4) one may solve such equations with the given Bj in class

.

Qm-j+a.

In addition one can also solve the Dirichlet problem for a wide class of equations which are not strol~gly elliptic. Puthermore, and this is perh;tps the most useful feature of the estin~ates, with their aid one may prove looal ~ ~ a r t u r b a t i theorellis o~~ for nonlinear elliptic squations. For example if FA(s u , DZmzc) = 0 is 21, nonlinear equation depending on a parameter 1 and
, ...,

data, and if the a first variation * of P at u, is a linear elliptic operator L whicll is invertible (i. e. for which the Dirichlet problen~ (7.1) has one aud ouly one solution) the11 for I R I sr~fficie~ltly small there exists a uuique solutiol~u~ of the n o ~ ~ l i ~ iequation ear wit11 zero Dirichlet data. The estimates also yield differentirtbilihy theorems a t tlre boundary for solutions of nonliuetrr elliptic equations. The estimates are derived followilrg the < 0 it is convenient to reaame the coordinates, set (xl ... xn-;) = x , x, = t , (2,, ,xn) = ( 8 , t) = P. We consider integral tmusforrns of functions f (2) iuto f u ~ l c t i o ~u(x, ~ s t) , t>O. Let K(x,t) be a kernel defiued ie the half space t>O aud l ~ o ~ ~ ~ o g e u e o t ~ s of degree 1- n

>

, ,

...

.

I

(I

here PI = x l2 t2)i/2; Asstinre that Q is colltiuuous on the half sphere PI = 1 , t 2 0 and assiltne also (this condition can be weakened considerably) that has continuot~s first derivatives OII tlre half splrere whioh, together with B itself, are bou~rded in absolute value by x . Ia additiou we make the basic sssumptiou

I

+

I

L'(x,O)d~,=o.

181-1

I

Here integration is over the unib sphere ( x = 1 of area. Consider the transformation u(x,t)=

I

K(x-y,t)f(y)dy

, with

d w, as element

diferential equations integratiol~ beilig over the elltire y space. Deuote tlre L, norm o f f by Ig, aud that of u ( x ,t) iu x , for ally fixed t , by I u l L p, t . THEOREX:1. For 0 < a 1

If

<

acheye c dqends only ow a rrtrd n . Here the novels vefev to the half space t 0 jbr u and the plane t = 0 for f. 2. For every t 0 and 1< p oo

>

,

>

< ,

where o depends only os P and n .

where c is an absolute oonstnst. Here (f)-112is defined in tavms of the Fourier transforvn o f f by

T(E)

There is at1 L, analog~ie of 3, which is however more complicated to state. We call Part 1 of the theorem a result of Privaloff type. I t is a simple extension of classictal results of Holder, Giraud and others, to wllicl~ it reduces if we set t = 0 . Part 2, a result of Riesz type, is a stnlighforwlttd extension of recent results of Ualtlero~~ a1111Zygmund [24], to which it reduces if we set t = 0 . For the special case of the HiIbert transfonn for ,n= 2 it is due to Riesz, a ~ ill ~ dfact it is proved by reduction to the Riesz result witlr the aid of a, device of [24]. Part 3, is proved with the aid of Fourier transfor~ns- one sl~owsthat the Fourier tri~llsfornl K ( 5 ,t) of K ( x ,t) with respect to the a variables is bou~~ded i l l absolnte value by constant (1 t I)-' from which the result follows easily. Part 3 plays an essential role in the derivation of the L, estimates. A

+1

,

Lecture VIII. The Boundary Value Problem in a Half Space; The P o i s s o ~Kernels. ~ I n this lecture we shell show how to solve explicity the elliptic system (7.1) with coasbant coefficiei~tsfor the special case of a half space. Making n slight change of llotation we sb@ll co~18iderthe space to be

L. Nrnlc~slcaa: On elliptic parNal

+

, ...,

n 1 itimeusional, mitli tlie first n coordinates denoted by a = (3, x,~) and tlle last coordinate by t . 111 the half spikce t 0 me cousider-for sim-

>

a

plicity the homoge~ieousequation, with D

where L is an elliptic operator of order 2 m with only highest order terms, satisfying the e coutlition 011 L B of the ~reviouslecture, i. e. for fixed real 5 =(l, tn) 0 the polyuolnial L ( E l z) lias exactly nb roots z on each side of the real axis. 011t = 0 we prescribe the derivatives

,... , +

I$-1,'= Gj(x) with the Qii in G r , for simplicity. Tlie solution w'ill be given i l l terms of kerliels Kj ( a , t ) j = l the Poisson kernels,

,

,...,m,

Kj(~-y,t)Qij(y)dy=2Kj*Qij,

(8.3)

where (: deuotes co~ivolution. Our constrrlction of the Ki is an exteusion of the coustructio~lgiveu by Agnion [25] in two dilneasions, n = 1 , but it is based OIL the Fritz Jehu ideutity (1.6) of Lecture 1 : For QJ (a) in Go"

where q is a non-negative integer of the stluie parity as s, A is the Laplacean, ant1 the priucipitl bnruch of the l~garithm is taken with the plane slit along the negative real exis. First some preliininaries. For fixed real t 0 denote by z t = z$ (6) k =1 m , the roots z with positive ilnaginary parts of L ( E l z) = 0 , and set

+

,...,

,

."

at

+

The coefficieuts are an~lyt,icill 6 for real 8 0 , a l ~ dIio~~iogeneous of degree p . With M+ we associate the polynomials (in z)

diflerential equations

The following relations are easily verified.

where y is a rectifiable Jordan coutour in the complex a plane enclosing all the roots r+(5) ill its interior; 8; is the Kroi~eckerdelta. We call uow writhe dowll the Poisson Kernels : For j - 1 2 a

for j-1<%

Here

and y is a Jordau c o ~ ~ t o uinr 3tit.z> 0 eiiclosiug ell the roots z of M + ( [ , a ) for all 1 61 = 1, l real. Before proving thiit these for~riulwsrepresent Poisso~~ kernels we observe, with the aid of the identities

(-

1y+r 2"~

(p+A)!(-l-p)!(2Y(

3

log -r = z p , pO

for 1,zc integers, and ~ 1 , ~sott1e ' apl)rnl)riate co~~stants, that sr may rel)rese~rt the fal~ctioi~s K, iir the forn~- with q a I I O I I - I I ~ ~ ~ integer L ~ ~ V ~ 11avi11g the same parity as n. -

L. NIBENBERG : On elliptic partial

44

where, for j - 1 2 n

,

and for j - l < n

+

I t is easily seen that Kj,, and all its derivatives up to order j q are contiuuous in the closed half space t 2 0 . We uom prove that the kel.aels Kj give11 by (8.7), (8.7)' are indeed Poissorr kernels. By iuspectior~ me see t l ~ a tthe Kj are analytic solutious of Lu = 0 for t 0 . H e l m a defined by (8.3) is ib solution. Setting

>

we shall show that uj beloltgs to Cw iu t 2 0 alrd that for t=O, k=1, Col~sitleriury pa,rt,ial derivative of ortler s of tcj q of the same parity as n , altd such that q 2 s -j

.

...,m.

Clloosing au iuteger

+ 2 we have, for t>O

after partial integration, recalli~igthat !Dj€ G.Since, as remarked iibove, D-j,, is continnous in the closed half space t 2 0 it follows that DSuj c m be exte~rdedas ;G contilrtious fiil~ctionill the entire closed l~alfspace t > O . Since s is arbitra,ry we have proved that uj E Cm iri t > 0 .

To verify (8.1 1) choose q snfficiently large so t l ~ a tq >j j =1 m Usillg (8.12) we have, for t = 0 ,

,..., .

after a change of variable. Assume first that k * j appropriate constants c', c"

. Usirlg (8.10)',

- 1c

+1 .

(8.10)" we find, for t = 0 , and

+.

by (8.6). Thus (8.11) is proved for k j Now suppose k =j If j - 1 n we have, usil~g(8.9) (8.10)' arid (8.6)) for some constant C'

.

- pj (jan i q!

>

eSdW6 I(y .E)q (log 7+ c/)/% ICI-1

M+

rj-l

Y

.

where yq (y) is a hornogeueous polynot~tislof degree q Similarly if j - 1 n we firlil, usirlg (8.10)" ilncl (8.6)

<

(8.14)"

D{-'

,

(y 0) = (- l)*lB'/ (n - j)!q!

(y

.

6)q

log

Y . E d q f yq (y)

l€l=l

where again y, dellotes a bomogeaeous poly~lo~uial of degree q : Prom (8.13), (8.14)', (8.14)" we fiud, after iliserting the value of (8.8), and rechanging variables, that

fij from

Here we have used the fact that

sil~ce ly, is a ~)olynombl of degree p and is therefore annihilated by ~ t + q ) ' ~ By . Joh1~7sideutity (8.4) the right side of (8.15) equtrls (Pi(%) , and the proof that the K , are Poisson kernels is co~nplete. We remark that the fiuletions K,,, are actually analytic ill t 2 0 except at the origin, and that for s >j q , D",,, is ho~noge~~eous o f degree j - 1 q - s . I t follows from onr proof above that if s = a 4 j- k > 0 then t K,,, ( x t ) < constant . x 12 t2)(1,+1)/2 l Di

+

+

, I

I

++

(I

+

Purlier~nore we see that becanse of tbe reprodnciug properties (8.11) of the K j we mHy ssqert that for x ~ / - lK j ( x

,t )g constant (I m +t 12

+0 ,

t2)(nS1)/2'

With the aid of (8.16), (8.16)' it is not diffioult to establish the following Extended Yaximuln Pi.ittciple : The so1ut;or~(8.3) of the Divichlet problem (8.1), (8.2) satisfies

where the least upper bounds are token with respect to all derivatives of ordev m - 1 and, 012 the left, with respect to all jx t) in the ha(f space, on the right with vespect to nll x . This is an a~ialogueof a ~pecialcase of Mirauda7e'extended nltixi~num prilrciple of [23].

,

BIBLIOGRAPHY [ I ] H. LEWY, A n ezample of a smooth liaear partial difetwtlial eqnation tuithout solution. Annals o f Math. 66 (1957) p. 155-158. 121 L. E H I ~ K N P R ESolutions IS, of some problems of division I, II. American Jourl~alo f Math 76 (1964) p. 883-903, 77 (19.55) p. 286-292. 7'he division problsni for dietributio*e, P~.oo. Nat. Aoad. Soi. 41-10 (1955) p. 756-758. [3] L. H ~ H M A N D EOn R , the theory of general partial differential operators. Aota Math. 94 (1965) p. 160-348. [4] B. MALGRANGE,Exietence et approximation des solvtions des dqaatiolls aux ddrirdes partiellee et den Cquatioas de convolntiors. Annales de L'lt~at. Fourier 6 (1955-6) p. 271-355. [ 5 ] F. T I ~ $ V E SSolutions , dlimei~laire d'lquations ~ I U Xdhivdea partiellea dPpead~ntd'un paramdtre, C. R. Aoad. Soi. P;rris 242 (1956) p. 1250-1252. [6] 1,. H ~ R M A N D ELocal I I , and glubal properties of futtdaete~ttal solutions. Math. Soandinavioa 6 (1957) p. 27-39. On the divieion oj distributions by polyaomiala. Arlciv. fiir Mnt. 3 Nu. 53 (1968) p. 555-568. [7] F. J O H N ,Plane waves and qherical meaas applied to partial differential uquations. Intereoionoe, New Pork, 1955. [8] N . DU I'~.rasrs, Some theorama about the Hieez fractional integral. Trans. Amer. Math. Soo. 80 (1955) p. 124-134. [9] L. N I I ~ R N B E I Il&marks G, on strowgly elliptic partial dvferetilial eqitations. Colnm. Pure Appl. Math. 8 (1955) p. 649-675. [ l o ] L. Bmns, L. h'IltENBERG, ( a ) On a repreee*tafion theorem for linear elliptic systems toitli diuoontinuous coefficients and its applications. (b) Oti linear and ronliaear elliptic boundary value problema in the plane. Aiti de Ct~nvegnoInter. sol. Eql~azionialle derivate Parziali, Trieste, 1954 (pnblished 1955). [ I l l DE GIOKGI,S~llla difJresziabilitlt e l'aaaliticitb delle estremali degli integrali mztllipli wgolari. Mem. della Aooad. delle Soienze di Torino. Ser. 3, V o l . 3 (1957) p. 25-43. [ l a ] J. NASH, Cantinuity of solution8 of parabolic and elliptic equations. A n ~ e r .Jonrn. Ma.th. SO (1958) p. 931-954 [I31 N. AI~ONSZAJN,On coercive integro differential qsadratic fornie. Conference on Partial Differential Equations, Univereily o f Kansas, 1954, Teohnioal Report No. 14, p. 94-106. [I41 8. AGMON, A. DOUGLIB,I.. N I B I C N B I C H IE~tinaaiee G aear the boundary for eotutiotts of elliptic partial differential eqtcntione eatisfying general bonadaiy conditions I. T o appear i n Comm. Pnre Appl. Mrth. [15] M. S C H R C H T I EIntegral I~, inequalities for partial differential operators and functions satiafying general boundary conditions. T o appear in C o n ~ mPnre . Appl. Math. V o l . 12, No. 1 (1969). [I61 8. AGMON, The coerciveness problem for integro diferential forms, Journal dlAnalyse Math. 6 (1968) p. 184-223. [17] L. H ~ ~ X A N D EOnR ,the interior regularity of the 8olutioh.8 of partial difereittial eqnationa. Comm. Pure Appl. ~ a t h 11 . (1958) p. 197-218. El81 B. MALGRANGE,Snr nne classe d'oplratetrre diffdrentiele hypoelliptiques. Bull. Soo. Math. Franoe. 85, 3 (1957) p. 283-306.

L. NIRPNBPRC: On elliptic [I91 ?. D. L A X , On Cauch,y1s problenb for l~yperbolic eq~rntiona and the differertiabilily of solutio~te of elliptic equations. Comlu. 1'11re Appl. Math. 8 (1955) p. 615 633. [20] L. BICI<S,Elliptic partial differential eqnatioss. Lecture notes o f t h e Senlinar it1 Applied Mathematics, University o f Colorado, June 1967. [21] L. S c a w a e ~ z ,Ecuaciones difetfel.enciales parcialee elipticas, L e c t ~ ~ r ests Bogota Colombia, 1956. [%4] L. H ~ ~ I ~ M A NOn D Pthe K ,regularity of tile soltilions of boundary problems. Aota Math 99 (1958) p. 225-264. 1231 C. M I R A N D A Teorema , del massimo modulo e teorema di eeisten~ae di unicitir per il problema di Dirichlet relativo alle equaziotii ellitliclie in due variabili. Alli~alidi Mitt. Pura ed Appl. Ser. 4, Vol. 46 (1958) p. 265-312. [a41 A. P. CAI.I)IERON, A. ZYGMUND,On singular integrals, Amer. Journal Math. 79 (1956) p. 289-309. [!5] S. AGMON,Mulli~de layer potetttials and the Diricl~let pt'oblem for higher order elliptic equations iqi the plane I. Comm. I'nre Appl. Math. 10 (1957) p. 179-239.

ADDITIONAL GENERAL BIBLIOGRAPHY [ A ] C. M I I ~ A N DEquazioni A, alle derivate parziali di tip0 ellittico, Springer, Berlin 1955. [ B ] l'vaaeactions of the Synbposit6m ot~ Partial Differealial Equqsntiotts. Berkeley California, 1955, published in Con~nl.Pnre Appl. Math., Vol. 9, No. 3 11966). [C] E. MAGICNEB, G. S T A M P A C C H I Aproblemi , a1 contorno pet- le equazioni dijfevenziali lineari di tip0 ellittico. Annitli dells Souola Norm. Snp. di Pisa Ser. 3, Vol. 12, Fasc. 3 (1958) p. 297-358.

I1:~tl.nt~to dngli A~rtiitli ilalltc Hc~colcc Nolsictle 8trpst~ior.odi

Pisa

Rel.ie 111. Vol. XIIJ. FHRO. IV (19.59)

THE 4,APPROACH TO THE DIEtIOHLET P R O H I J E M ~ )

PARTI REGULARITY THEOREMS

1. Introduction.

In this paper we present a L, approach to the Dirichlet problem and to related regularity problems for higher order elliptic equations. Although this approach is not as simple as the well known Hilbert space approach developed by Vishik [32] ~ i r d i [14], n ~ Browder [6 ; 71, Priedrichs [12], Morrey [22], Nirenberg [23], Lions [18] and others, it has the advantage of a greater generality. Thus, for example, we shall be able to treat the non-homogeneous Dirichlet problem in a much more general situation not restricted to solutions having a finite Dirichlet integral (in this connection see Magenes-Stampacchia [19, $ 91 and the recent paper of Miranda [20]). The method is also applicable to elliptic operators which are not necessarily strongly elliptic. We remark further that the same method could be used to solve a general class of boundary value problems. This will be done in a subsequent paper where me shall also derive Lp integral inequalities for a system of differential operators acting on functions satisfying general boundary conditions, simular to the a coercive 9 L2 inequalities derived by Aronszajn [4] Agmon [2] and Schechter [25]. Recently Schechter 126 ; 271 presented a Hilbert space approach to general boundary value problems including the Dirichlet problem for non-strongly elliptic equations. His method is based on the L, estimates of Agmon-

(') Presented in part (for p = 2) at the intertiational conference on partial differential equations organized by the C. I. M. E in Pisa, September 1-10, 1958. Sponsored in part by the Office of Scientific Research of the A. R . D. C., U . S. Air Force, through its European Oftice, under Contract Nu. AF 61 (032)-187.

SHMUNI.AGMON: The Lp nppronch to

Douglis-Nirenberg [3] (see also [2; 251) and on known L, regularity theorems. Our L, method which utilizes new regularity theorems is quite different and the results we obtain are stronger in various respects (I).Other existence results utilizing the continuity method were given by ~ ~ r n o n - l ) o u ~ l i s - N i renberg [3]. The Lp approach to the Dirichlet problem is based on a Lp regularity theory for very weak solutions of the Dirichlet problem. To obtain such a regularity theory we use some of the ideas gf a method originally devised by Nirenberg [23] with the following essential modifioation : instead of using 3 Garding's inequality we use the explicit ~olutionof the Dirichlet problem for elliptic operators with constant coeffioients in a half.space, and the L, estimates for such solutions derived in [3]. The paper is divided into two parts. In Part I we give the basic regularity theory, both in the interior and at the boundary. This part has an independent interest and entails most of the work. We remark that when we consider the simpler problem of interior regularity me consider also weak solutions of overdetermined elliptic systems and derive Lp estimates for such solutions. In Part I1 we shall combine the regularity theory with some general results on Ranach spaces (using in particular a result of Fichera [lo]) to develop the Lp existence theory for the Dirichlet problem.

2. Notations and defil~itions.

Throughout the paper we denote by G a bounded domain in n dimensional space with boundary 8 G and closure % We denote by x =(x, xn)

,...,

...

1 -

the generic point in the space and put I x I = (xy f $8;)'. We sa,y that ff is of class Cj if with every point x0 = (x; ,xi)E d G one can associate a sphere S having its center in zO such that d G S admits a representation of the form :

,...

for a suitable k ; g being a function defined in some neighborhood U of (xy xi-, a:+, x:) possessing there continuous derivatives up to the order j.

, ... ,

,

, ... ,

(1) For instance, Schechter's method is applicable only to snch problems for which tho sol~~tiotr 'of the acljoint probleltt is unique, whereas \re get the alternative in the generd case without :my uniqueness assnmptiou

the Dirichlet problem

Similarly, G is said to be of classe COJif around each of its points the boundary JG admits a local representation of the form (2.1) with a function g satisfying a Lipschitz Condition in the neighborhood U Finally, G is said to possess the cone property if every point in is a vertex of a closed right spherical cone of fixed opening and height which belongs to 6. I t is rea,dily seen that if G is of classe Cog1 then it also has the cone property. We shall denote by Ck(G) (resp. Ck the class of complex valued functions possessing continuous derivatives up to ther order k (0 < k 5: oo) in G (resp. 6). The class of infinitely differntiable functions with compact support in G will be denoted by C r (G). Let, now, j be a non-negative integer and p a real number 2 1. For a function zc (x) belonging to Cj (3)define the norm :

.

a

(0)

where here and in the following a stands for the multi-index ( a , , I a = a, a,, and Da is the partial derivative :

I

+ ... -+

..., a , , ) ,

We also put : d ni = dX$

(i = 1

, ..., n)

and

D = (D,

, ... , D,)

The linear space Ci (Cf) is clearly not complete under the norm (2.2). Completing it we obtain a Banach space which we denote by Hj,Lp(G). We retain the notation 11 I l j , ~ p ( ( t , for the norm in H ~ , L ~ ( The G ) . space HO,Lp(G) is simply 'the space Lp (G) and we shall usually write 11 IJLp(G) for the norm in this space. The classes of functions Hj,Lp(G) were investigated by many authors (Sobolev [30], Morrey [21], Friedrichs [Ill, Stampacchia [31], Deny and Lions [g], Gagliardo 1131 and others). Some of the properties of these classes will be described in the next section. Here we limit ourselves to some remarks. By the identification mapping we can consider Hj,Lp(G) as a linear subset of L, (G). A function u E HjVLp ( G ) mill possess generalized derivatives up to the order j which we term strong Lp derivatives. To define these let

Se~uer.AGMON : The Lp approach to (ui]Ll be a sequence of f~~nctions in, Cj (Cf) such that

Then, there exist functions ua E L, (G) (0 lim (-.m

11 Da ~i - ua IIL~(G)

I cl I <.j)

such that

=0 .

The functions ua are by definition the strong Lp derivatives Dau of u in G. They are uniquely defined. We shall say that a function u belong locally to Hj,Lpin G - writing u E H$" (G) - if for every 8 E G there exists a sphere SC G with center at x such that u E HjJLp(S). I t is readily seen (using a partition of unity) that if u E (G)then u E Hjtq ( G I )for every domain Gt such that Cf, C G I n connection with the Dirichlet problem we shall have to consider the subclass of functions in Hj,Lpwhich together with some of their derivatives, vanish at the boundary in a generalized sense. To make this more precise suppose thitt G is of class COJ. Let u E (G). Then, as it is well known, one can define for such u its trace y (u) on the boundary. For instance, one can use the following procedure. For u E Ci ((f) y (u) is simply the restriction of u on dG. In this case it is easily established that

HE;

.

-

with a consta,nt which is independent of u. Hence, u y (u) is a bounded linew transformation from C1 @) (considered as a subset in HI,%(G)) into Lp (GG). Since C1((f) is dense in H ~ , ( LG )~one can extend the transformation in a unique manner by continuity to the whole of (G). This defines the trace on the boundary of a function uE H l , ~(G) p as an element of L,(dG). Let, now, m j be positive integers such that m j. We denote by (G ; (Da)lalsm-l) the class of functions u E Hi,Lp (G) which satisfy the boundary conditions

<

,

Dau=O

on dG for O < l a l < m - 1 ,

where (2.4) is taken in the sense that y (Da u) = 0 as an element of Lp @a).

t8e DirioAlet problem

We observe that H j , ~(G p ; (Da)lalSmn-l) is a closed snbspace of B j , ~(G), , and that a function u belonging to hi,^, ( G ; ( D a ] l a l ~ mn- lCw"-' ) satisfies the boundary conditions (2.4) pointwise in the ordinary sense.

(6

3. Calculus and properties of tho clrtsses Hj,Lp.

We have remarked already that a function belonging to Hj,Lp(G) possesses strong Lp derivates up to the order j in G. Considering such a function u as a distribution in G (Schwartz [28]), it is readily seen that the strong Lp derivatives are also the distribntion 'derivatives of u which are thus functions belonging to L,(G). I t is very convenient that under general con. ditions on the domain G one can reverse this statement. We have: THEOREN3.1. Suppose that G is of class COJ.Let u E Lp( G )( p 2 1 ) and msume that the distributioe derivatives of zc of order ( j are functions belonging to Lp(G). I. e., assume that there exist fultctions ua(x)E Lp (G), 0 1 cc 15j

<

(weak derivatives in the terminology of Friedriclts) such that

for all y E (G). Then, u E H j ( G ) and its distribution derivatives ua coincide ' 9 with its strong Lp derivatives Dau ( 1 cc I ( j ) . The weaker conclusion that u E ( G ) is well known and Iras esta. blished by various authors (Friedrichs [ll], Sobolev [30], Deny-Lions [9]). The theorem as stated is due to Gagliardo (*) [13]. For more regular domains it mas established by Babich [b]. The folloming remarks are obvious. If u E Hj,,@ (G) and a E Cj (@), then v = au belongs to H j , ~(,G ) and the strong derivatives of v are obtained by the standard T~eibnizrule. If, moreover, O is of class 0021 then the boundary values y (Dav) (1 a 1 5j - 1) are obtained by the same rules. The classes E j j L pare preserved by homeomorphism of class 12.That is, let s" x (x") be a one to one mapping of onto G such that the mapping and its inverse possess continuous derivatives up to the order j in the corresponding closed domains. Then the mapping u uX, u" (m") = u (z (n*)), is a homeo-

HE;

-

a*

-

(2) It should be pointed out that Gagliardo i~ not using the notion of a weak derivative but a different notion which is, however, equivalent to it. Also, the proof of the main approximation theoretn [13 ; p. 1121 could 1,e repeat,etl mortl by word for fi~nctions pusserrsing weak derivative8 in the sense of (3.1).

SHMUEI. AGMON : The Lp approach to

morphism between H,,L~.(G)and Hj,hp(Ox). Also, to compute the strong derivatives of u* one applies the usual chain rule. The same remark applies to the trace a t the boundary of derivatives of order sj- 1 when G is of cl&ss COJ Most of the following lemmas are the Lp modified versions of the calculus L, lemmas given in Nirenberg [23]. Unless otherwise stated we shall assume in these lemmas that G is of class COJ. LEMMA3.1. Let u E Lp (G),p 1. #uppose that k is a weak limit in Lp of a sequence of &unctions (N)which belong to H j V L p ( fand f ) possess unijormly bounded norms uk Ilj,Lp(~) Then, u E HjqL;(G) and its derivatives of order < j are the weak Lp limits of the corresponding derivatives of the jicnctions uk. Proof: From the weak compactess of the unit sphere in Lp ( p 1) it follows that there there exists a subsequence uk, such that Daukr converges weakly in Lp to a function ua E Lp (a), 1 a 5j Hence, for every function Y EG ( G ) :

.

>

11

.

>

I

I

G

=1 k'-Co

(- 1

9 ua d,x = lim k'-m

.

I

p, Daukldx

Q

I

.

I

Da y ukrdx = (- l)lal Da p, . u dx. G

Q

Thus, ua is the distribution (weak) derivative Dau. But then, since uaE Lp (G), it follows from Theorem 3.1 that u E Hj,Lp(G) and that u@coincides with the strong Lp derivative Dau. Moreover, from the uniqueness of the derivatives it follows that the whole sequence D a q converges weakly to Da u anand not only a subsequence. Using Theorem 3.1 one also obtains readily the following LENMA 3.2. Suppose that u belongs to HjILp(G) and that its j' th order derivatives belong to Hk,Lp(O) then u E H j + k , ~(G). ~ NOTATION : Let h = (h, h,,) be a real non-vanishing vector. We shall use the symbol dh to denote the difference quotient operator:

,

,...,

,

LEMMA3.3. Let u E Hj,Lp(G) ( j2 0 p > 1). ~Yupposcthat there exists a constant C such that for every subdomain G,; G, c G :

the Diriohlst problettb

for all suficently small vectors h. Tile% $6 E Hj+l,Lp(G) and

Proof: Consider first the case j = 0 . From (3.3) and t,he weak compactness of the unit sphere in I;, it follows that there exists a sequence of vectors (h7m)z-,in the direction of the ~6 axis, hm 0 , such that the sequence dnm u (wt sufficiently large) tends weakly in Lp (GI) to a function ui; and this in every fixed subdomain G, G, c G. Since 11 ui ( C for all such subdomains, it follows further that ui € Lp (17). Now, from the definition of weak convergence we find that for all fun-, ctions g, E C r (G):

-

I~L~(G,)

,

S

g, ui dx = lim m-m

G

S

g, .

u dx

G

m+w

This shows that ui is the distribution derivative Di u in G. Since Diu € l;,(G) (i = 1 n) me conclude from Theorem 3.1 that u E (C). Clearly, we also have I1 Di u IIL~(G! I c .

, ...,

Next, assume that j 2 1. Let again (h"] be a sequence of vectors in the direction of xi tending to zero. I t is easily seen that a,, u converges to Di u in Lp (G,). Assuming without loss of generality that G, is of class COJ and applying Lemma 3.1 to the sequence Id,,, u ] ,it follows that Diu E Hj,Lp(G,) and that 11 Di t~ l l j , ~ p ~5~ , c~ (i = 1 , n )

,

From this and from Lemma 3.2 we conclude that u E Hj+l,Lp(GI) for any subdomain Gi of class CoJ (and consequently for any subdomain G, G, C G). Since all the distribution derivatives of u of order (j 1 are functions belonging to Lp (G) it follows from Theorem 3.1 that u E Hj+l,Lp(G). That (3.4) holds is obvious. By the same argument used to prove Lemma 3.3 for j = 0 one obtains R, x, 0 Let t~ LEXMA3.3'. Denote by ZR the hemisphere I a ( be a function beionging to Lp (ZR),p 1. #uppose that there e ~ i s t sa coastant C such that for every R' R:

+

<

>

<

,

> .

S H M U ~AGMON I. : The Jp approicch to

,...

for all suficiently small vectors h of the form h = (hi , h,-, ,0). Then the n - 1 are functions belonging to distribution derivatives Diu for i = 1 , L p (ZR)with 11 Di u 5 0. The following known lemma will bc useful. LEMMA3.4. Suppose that G has the cone property. Then, for all functions u E Hj,Lp( G )( j 2 1) and every e > 0 the follou~ing inequality holds :

...,

,

where C is a constant depeuding only on e ,j p and G. Lemma 3.4 for somewhat more regular domains was established by Nirenberg [24](5). The inequality for domains which have t,he cone property was proved by Gagliardo [13]. Finally, we conclude this section with the well known integral inequalities of Sobolev [30]. THEOREM3.2. Suppose that G has the cone property. Then the functions u belonging to ( p > 1) satisfy the following relations. n 1 1 then uE L, ( G ) where q is de$ned by - = - - j Also, (i) If p < 9 q p n

.

with a constant depending only on n ,j ,p and Q. 'n

(ii) If p == : then u EL, (G) for every 1 9

(iii) If p

n >then u is a continuous function

null set) such that (3.6)'

< q < co and

j

SUP I u I 5 Const. a

11 u I(~,L~(G)

(3.6) holds.

(after correction on a

7

with the same constant dependence as above. REMARK:If the boundary of the domain is somewhat more regular, e. g. if G is of class 0081, one can assert in. case (iii) of the theorem that u satisfies a Holder condition in Cf. 4. Some lemmas related to elliptic operators with eonstat~tcoefficients.

Let A ( x , D) be a linear differential operator with complex coefficients operating on functions u (x) defined in a domain of En;,. Denote by A' the

(3)

The analogous one dimensional case is due to Halperin and Pitt.

the Dirichlet problem

leading part of A, i. e. the part of highest order terms. A is said to be elliptic in the domain if for every point x in the domain the characteristic form A' (x 8) 0 for all real vectors 8 = (E, 5,) =!= 0. I t is well known that if n 3 and A is elliptic then its order is even. This is not necessarily true for n = 2. In this section we shall consider an elliptic operator A of even order 2m with constant coefficients and with no lower order terms :

, +

, ... ,

>

A ( D ) = 2 a , Da lal-2m

A being elliptic there exists a constant 1

.

> 1 such that

for all real vectors 8. We term 1 the ellipticity constant of A . We denote by xf= (xi , x , - ~ ) the generic point in En-,and whenever convenient write x in the form (a', x,). We also put, D,,= (Ill Dm-,) and D = ( D x f D,) Write the operator (4.1) in the form A (D,, , D,). For a fixed real vector E' = (5, , En+) 0 consider the roots (in &J of the polynomial d (5', 5 , ) . If n 3 the ellipticity of A implies the exactly half the roots possess a positive imaginary part (see [3]). This is not necessarily true for n = 2 if the coefficients are not real. In general we shall say that A satisfies the e roots condition H if for every fixed real vector 5' 0 the polynomial d fn) has exactly m roots with a positive imaginary part. The following two lemmas are basic for the proof of regularity in Lp of weak solutions of elliptic equations. The 6rst rather known lemma will be used to establish interior regularity (and Lp estimates) of weak soliltions of elliptic equations and overdetermined elliptic systems. The second lemma will be used to establish regularity at the boundary of weak solutions of the Dirichlet problem. In both lemmas A will stand for the elliptic operator (4.1) and p will denote a number 1. In Lemma 4.2 we shall assume in addition, if n = 2 , that A satisfies the e roots condition ,> introduced above. We shall denote by SE the sphere I x R and by 2Bthe half sphere Ixl O . LEXNA 4.1. Given a function f E C r (SR) there e ~ i s t sa function vE Cm (&) such that

... ,

, ... ,

, . ..., +

>

+

(r,

>

I<

-

(4.2)

Av =f

in Sn

and (4.2)'

, ,

where C is some constant depending only on n nz p ,Id and 1 (but not on f or v). LEMMA4.2. Given a function f E (zR) there exists a function v € C" ( y ~ ) such that

(4.3) and

I

Av = f

in

D&=0

for

-

ZR xn=O

11

(4.3)'

(lxl
5

I~~%L~(zR)

j=O

,...,m - 1 ,

11 f 1lLP(28) 1

where C is some constant depindig only on n , m , p , R and 1.

To establish ~ e m b a4.1 we simply define

where F (2: - y) is a suitable chosen fundamental solution of A with pole ilt x = y I t is well known (e. g. P. John [16]) that there exists a fundamental solution having the form :

.

I? (2)= I 5 12rn-n y -

(,:I)

+P

(8)log

I $1,

where P (a) is a polynomial of degree 2nz - n if n is even, 2m 2 n , and P (x) is zero otherwise ; y (y) is an analytic function clefined on 1 y 1 1 From (4.5) it follows that l=

for ( i ) I u I )0

.

, in

case n is odd or n is even and greater than 2m ; (ii) if n is even and not greater than 2m. If n is even and O
I a I > 2m - n

IDa I? (8) (

< Const. I x 12m-n-lal

+1

(1

1% I x

11).

Inspection of the explicit formulas for the fundamental solution (in [lB]) shows that the constants in (4.6) and (4.6)' depend only on m , n , l and

the Din'chlet problem

I I

I I.

a Furthermore, it is easily established that Da P (z) for u = 2m is a homogeneous function of degree - lz with a zero mean on the sphere IxI=l. Choosing a proper normalization of P, the function v defined by (4.4) is infinitely differentiable and satisfies (4.2). Furthermore, from the properties of the fundamental solution mentioned above and from the well known theorem of Oalderon and Zygmnnd 181 on convolution transforms with singular kernels, it follows readily that

, , ,

where C is a constant dependig only lz m p R and 1. Hence, the function v defined by (4.4) answers all the requirements of Lemma 4.1. The proof Lemma 4.2 is more involved and depends on the solution of the Dirichlet problem for A in a half space and related L, estimates. We shall denote by E,+ the half space xn 0 . In its simplest form the Dirichlet problem for A in E: is the following PROBLEM : Given functions q i(2') qm(xl), in$nitely aiferentia.ble and of compact support in En:,_, , jind an in$v~itely dierentiable function u (xl,xn) in. B i such that

>

,... ,

(x-lu(XI, 0)= q,(xl) for j = 1 ,..., m. This problem (a special case among a whole class of boundary value problems) was solved in [3](4), where it was shown that there exist kernels Kj (x', x"~)(j= 1 nz), defined and infinitely differentiable in 2; except for the origin, such that a solution of (4.7) is given by the formula:

, ...,

We mention the following properties of the kernels Ej also established 131. Let q be a nomnegative integer having the same parity as n - 1. The kernel Kj admits a representation of the form

4 (x', xn)= A;,1(n-l+q) qjq (s',an) (4)

For s = 2 and A real the solntion was given in [I].

SHMLIEI. AQMON: The Lp approach to n-1

where A,, is the Laplacean 2 D:, and Ej,,are certain kernels which are infinitely i-1

differentiable in 5 : except for the origin which, moreover, satisfy the following inequalities in E::

I D a q,,(x) I (Const. 1 x 1 j-l+q-lal

(4.10)

for I a I < j - 1 + q ,

(1

+ 1 log I 311)

and

+ ,

for I a I Lj q where the constants in (4.10) and (4.10)' depend only on n m ,q I a I and the ellipticity constant A . Let, now, w (x', xn) be an infinitely differentiable function with compact support in 2:. By the preceding a solution u E Cm (1:)of the Dirichlet problem

,

,

(4.11)

Au=O

,

;I)'-'u (8' 0) = &-'

20

in

,

Hi,

,... ,m ,

(8' 0) for j = 1

is given by the formula

Moreover! we have LENMA 4.3. The solution u satisfies tile following inequality i n L,,p>

1:

where c, is a conetant dependilzg only on m , n , p and 1. If,i n a,ddition, the support of u1 is contained i n tile halfsphere ER then

where 0, is a constant depending only on n ,m ,p ,l and R. Lemma 4.3. was proved (essentially) in [3] (compare also Koselev I171 for the Lp estimates involved). For the sake of completeness we shall present a. somewhat simplified version of the proof later on. I t is with the aid of this lemma that we shall now give the

the Diriahlet problem

Proof of Lemma 4.2. Extend the function f (x) (f E C)~(ER)) as zero OU. tside Z I I . Denote by some fixed infinitely differentiable function such that 1'R = 1 for I x I R Cfi zz 0 for I x I 2 2R. Define :

cfi < ,

where I? is the fundamental solution of A introduced before. Ulearly, w is infinitely differentiable, w = 0 for I x I 2R, and

>

Aw=f

for I m l < R ,

II W I I Z ~ , L ~ ( X 1 ~ I ~Qi) I1 f llLp(XR)

7

,

where C , is a constant depending only on n , m , p l and R. Let, now, u be the solution of the Dirichlet problem (4.11) given by (4.11)' with w defined by (4.13). Put:

Then, v has all the properties required by Lemma 4.2. Indeed, v E C" By (4.14) and (4.11) :

I

Av = Aw =f for x E

(2:).

-

BE,

D ~ - ~ V = D ! - ~ ~ O - - D ~ - ~for ~ = xOn = O , j = l ,

...,m .

Fi~lally,using the estimate (4.12)' of Lemma 4.3 and (4.14), we get

,

, , ,

where 0, C are constants depending only on n m p l and IC. This estitblishes the lemma. We shall conclude the section with a proof of Lemma 4.3 based on the properties of the kernels Kj mentioned before. We shall need first the following SUBLEMMA : Let G (x) = G (0' x,J be a kernel: cle,/i~tedand measura,bIe in the half space E$ suck that

,

Sa~urr.AGMON: The L, approach to

, > 1 , consider

for so?)be constant G. For v € ELp(E:) p

the transform

Then, u. 6 Lp (E:) and

where y is a constant depending only on n and p. Proof: Set \dl(x)= for x n > O ,

I M ( x ) = - ~ x l - ~for

Extend v as zero for.

xn< 0 .

.

c,<_ 0 Then, for xn > 0 :

8%

-

Now, M ( x ) is an odd homogeneous kernel of degree n bounded on I x 1 = 1. Eence, we are in a position to apply the Calderon-Zygmund theorem [8] to the last integral (4.17), from which it follows readily that

y depending only on n and p. This proves the sublemma.

To prove Lemma 4.3 we shall first transform formula (4.11)'. To this end note that (integrating by parts with respect to yn)

i

0:-'w (y', 0 ) .Kj (x' - y'

En-1

,Xn)

dy'

the Diriohlet problem

where here and in the following all differential operators under the integral sign act on the y variable unless otherwise indicated by a subscript. Summing (4.18) over j = 1 m me obtain for the solution u of (4.11) the representation :

, ... ,

where (4.20)

Using (4.19) and (4.9) we observe that if q is a non-negative integer having the same parity as w - 1 then

where

ZjBq are

kernels given by

From (4.20)' it is readily seen that the inequalities (4.10)-(4.10)', satisfied by the kernels K,,, , are also satisfied by the kernels Put :

&,, .

1n

so that by (4.19) u = 2 u j . To establish the lemma it will suffice to show 0

that the inequalities (4.12) and (4.12)' hold for u j . We shall prove this for j odd. The proof for j even is similar.

SHMUICI. AGMON: The Lp approach to

+ + .

Choose q = 21th n 1 From (4.22) and (4.21) we obtain after obvious integration by parts with respect to y' :

Differentiating (4.23) ye thus obtain :

Suppose, firat, that I u I = 2 m . Using the estimates (4.10)-(4.10)' which, as was pointed out before, are also satisfied by the kernels Zj,, (q=2mf n+l), we find that

with a constant G depending only on n , nb and A. Thus, applying the Sublemma to a typical integral of (4.23)' it follows readily that

where y, depends only on n and p. This yields (4.12). Suppose, now, that 0 (( u 1 2m - 1 Prom (4.10)-(4.10)' one finds readily that in this case

.

I a A,,

1 . 7 (3-1)

6, (x', zn) I < Const. ( I x

Da

(-(n-l)

+1 x

1Zm)

the DirieAlet problent

,

with a constant depending only on n nz and 1. If, furthermore, the support of w is contained in it follows easily from (4.23) and (4.25) that for IuI 2nb:

zB

<

.

where C, is a constant depending only on n , wz, 1,p and R Since by the preceding (4.26) (with a suitable constant) holds also for I a = 2m we conclude that the functions uj (and consequently u) satisfy ($12)'. This completes the proof of Lemma 4.3.

I

5. Preliminary regularity lemmas. In this section we begin with the discussion of the regularity problems of weak solutions of elliptic equations in the framework of the Lp theory. We shall discilss both the problem of interior regularity (also for weak solutions of overdetermined elliptic systems), and the problem of regula,rity a t the boundary for weak solutions of the Dirichlet problem, We consider a linear elliptic differential operator A of order 2m (variable complex coefficients) defined in Q:

We denote by A' the leading part of A and by A some consta.nt 2 1 (el lipticity constant) such that

for all real vectors and x E g . We introduce the following DEPINITION 5.1. The coeflcients of A will be said to satisfy Condition 1j; K ) (in Q) j being a positive integer and K 0 if

> ,

,

a, E C lal+j-2m(8) for

1 a 1 > 2m -j ,

tokerea,s the remaining coefjcients are measurable boufided functions in G . (ii) The fol2occ;ing inequalities hold in ($ : and

I D P a a l ~ Kfor

lul>2m-j,

la,l
for

IbI
Iul<2m-j.

SHMUKI.AGMON : The Lp approach to

We also introduce the scalar product notation ( f , g)G to be used from now on throughout the paper :

Here f and g are two functions defined in G such that the integral (5.2) makes sense. In this and in the following section the domains of definition G mill be either the sphere SR ( I z 1 R) or the hemisphere 23 ( I s I R I 8, O ) , We shall denote by 8,Z m the, part of the boundary d B E situated on the hyperplane 3, = 0 and by, d, B R the part of d 2~ situated on I z 1 = R (8 & = 8,Z R u 8, ZR). We also recall that by 8h we denote the difference quotient operator :

<

<

>

,

...

I& = (h, , , h,) being a real vector f 0 . We shall now state two basic regularity lemmas. LEMXA 5.1. Let A be an ell&tic diferential operator of order 2m de9ned i n S R , with coeflcients satisfying Condition ( 1 ;E]Let, further, u be a function belonging to L, ( S R ) ,p > I , such that for all functions p E (SR) the following inequality holds :

.

1

1

,

where p' is the esponent conjugate to p : -$ - = 1 and C is a constant. P P' Then, there exisfs a positi~e number ro < R and a constant co such that

.

,

for all suficiently small vectors h Both r, and c, depend only on n , 7n,p R , K and the ellipticity constawt 1. LEMMA5.2. Let A ( 3 , D ) be an elliptic diferential operator of order 2nb defined i n yR,with coeflcients satisf?jing Condition ( 1 ; K ) I f n = 2 assume also that A' (0 D ) satisjies the ci roots condition introduced in § 4. Let, further, u be a function belonging to Lp(zR), p 1 , such that

,

>

,

.

the Diriolblet problem

for all functions q E Cm (%a) ~atisfying the bounddar?/co~zdjtions:

Iq

0 in a neighborhood of d , 2 ~ .

<

Then, there esist a positive nomber r, R and a constant coyhaving the same dependence as in Lemma 5.1, such that

.

for all sufjciently s?)tall vectors h parollel to the hyperplane xq,= 0 Proof of Lemma 5.1 and Lemma 5.2 : We shall prove both lemmas at the same time. In the sequel a, will denote the sphere 8,. in the case of Lemma 5.1, and the hemisphre 2, in -the case of Lemma 5.2. By our assumption the .coefficients of A are measurable functions bounded by K in o x . Moreover, the highest order coefficients possess tirst order derivatives also bounded by K. Without loss of generality we may assume in the following that A contains no lower order terms. Indeed in the general case let A = A' A" where A' is the leading part and A" contains only terms of order < 2m - 1. By Holder's inequality

+

Consequently the operator A could be replaced by A' conditions of the lemmas with C replaced by

which satisfies the

Let r be a, positive number
and extend zc, as zero outside GB. Let, further, v be and arbitrary function belonging to Cm(&) which in the case of Lemma 5.2 (a, = 2,)also satides the boundary conditions (5.9)

D$v=O We have :

for x,,=O,

j=0,

... , m - 1

SHBIUEL AGMON : The Lp approaol, to

where B is a linear differential operator of order Bm - 1 with coefficients which are linear combinations in a, Dp [, Hence, making use of Hijlder's inequality, we obtain

.

, ,...,

where here and in the following ce cs denote constants depending only on n , m , p , R , K , 1, and r . Consider now the function = t,.v extended as zero outside &. E C r (SR) in the case of Lemma 5.1, while y E Cw ( j ~and ) satisfies the boundary conditions (5.6) in the case of Lemma 5.2. Hence, applying the inequality (5.3) in the first case, and the inequality (6.5) in the second, we conclude that

Combining (5.11) with (5.10)' we thus get :

(02

for all functions v E Cm which in the case of Lemma 5.2 also satisfy the boundary conditions (5.9). Next, let I& be an arbitrary vector such that 0 I h I 976. In the case of Lemma 5.2 h is restricted, in addition, to be of the form h=(h, hW1 0 ) . Define the function

<

,

fh

Then,

fh E Apt

(4= ( d h ur ( p - l sign

<

,...,

( 8 h ur) (5).

(04,

and the support of fh is contained in with support in OR/B such that

OB/3.

Let, further, y h be a Cw function

(F) As nsnal : sign z = - if z $. 0 , sign 0 = 0 IzI 2

.

.

the BDim'cklet problem

We now make use of the lammas established in 4 4. Thus, in the case of Lemma 5.1 apply Lemma 4.1 to the elliptic operator AO= A (0 , D) functiou f =ji and exponent p' . There exists by the lemma a function V h E Cm (RK) Such that

-

,

,

.

where y is a constant depeuding only o n vz., m p , R and 1 Similarly, in the case of Lemma 5.2, applying Lemma 4.2 it follows that there exists a function uh E COD such that

(xE)

and (5.16)' where y is a constallt having the sitme dependence as above. Using (5.13)' and (5.14) me observe that in both cases

.

Consider the function 8-h v h I t is a well defined infinitely differentiable function in 0 . , Moreover, in the case of Lemma 5.2 it also satisfies the vh we have : boundary conditions (5.9). Hence, applying (5.12) to v =

Also, one clleoks easily that

where N is a constant depending only on n. (one can actually take AT=%). Combining (5.18), (5.19) and (5.17) we thus get

S ~ ~ u s AQMON r, : The Lp approach to

,

Using the. relation f (fg) - d-h f .g (a- h) and noting that tbe g= support of u, is contained in &B while I h I r/6, we readily obtain :

<

Since the coefficients of Ah are bounded by n K we have (using 5.17):

I(ur ,A-1

(5.22)

I K 11 %-

VL (8 - h))sr

/(+(a,)

11 vh I l % m , ~ ~ c5 (a~)

Thus, combining (5.20), (5.21) and (5.22), we get (5.23)

1 (dh '%

A vh),

I < I(

~9 rA

I + I (ur

(d-h vh))~,

A-h vh (8- h)),

Write

Using (5.15) (resp. (5.16)) we have: (5.25)

(bh '% .A0%)ap= (dh ur

-

fh)ar

= (dh

,f&,+

By (5.13) : (5.25)'

( a h Ur 7

h)or= 1 8h Ur /(:p(orl

Also, from Holder's inequality and (5.14) we get

.

( a h Ur

,.g-

fh)aT

I<

the Dirichlet problem

Thus, combining (5.25), (5.25)' and (5.25)" we obtain

I (dh MT

(5.26)

A0 vh)a, I >

Now, it follows from our assumption that the coefflcients of A -A0 me bounded by n E r in a,. From this and from (5.17) we get, using once more Holder's inequality :

We shall tix now r choosing

With this choice of r we obbain from (5.27) and (5.26): (5.29)

, A ah), I 2 1 (88 , A0 vh)ar 1 -

1 (dh

Ur

Finally, from (5.23) and (5.29) we get

If we now choose r

I x I (ro, I h I < r,, , we

r -, co = 3 c, , and 6

O -

note that

ah

zl,

= dh zl for

obtain from (5.30) :

for all h sufficiently small (h parallel to This establishes the Iemmas.

x,,= 0 in the case of Lemma

5.2).

Lemma 5.1 and Lemma 5.2 yield (respectively1 the following corollaries. COROLLARY5.1. Suppose thut the conditions of Lemma 5.1 hold. Then, uE~:y ( SiB )~ . dloreoner, for every R' R the following inequality lbolds :

<

,

, ,

where c, i s a constant depending only on m n , p l , K , R and R'. COROLLARY5.2. Suppose that the conditions of Lemma 5.2. hold. Then, for every R' R the distribution deriaatives I)i u fbr i = 1, ,n - 1, are functions belonging to Lp (.ZRt) R Z L C ~ Lthat

<

...

where c, is a cnnstant having the s m ~ sdepewdence as a,bove. To prove Corollary 5.1 let d = R - R' and denote by Sa~,,the sphere Ij 8 - $0 r Apply Ifemma 5.1 to cc in &,n after ohvions translation of variable), a0 E &, From the lemma, combined with Lemma 3.3, it follo~vs that there exist positive conatants r, d and c,, both ro and co depend only on n 9 ) ~ p t , K , R and d l such that u E H 1 , ~ ,(Sd,,) p and

< .

'

.

, , ,

<

EL,),

Covering SR, by a finite n n m b r of spheres S@,,, (xO using (5.33), me conclude that U E H,,Lp(SRt) and that, furthermore, the inequality (5.31) holds. Corollary 5.2 follows similarly from Lem~ns5.2, Lemma 3.3' and Corollary 5.1. ( 6 ) .

6. The brisic regoli~ritytheoretns.

We pass to the main regularity results in tlre fi-nmework of the Lp theory. The first theorem dealing with interior regularity is the following - THEOREN 6.1. Let A be an elliptic operator of order 2 rn [7) depned i n S R , with coefficients sati8fying Condition { j ; K ) ,j being a n integer such that

It shonld be ohervetl here that in the 'exceptiona1 case n = 2 the operator satisties the ((roots conditions for every 8'1E This follows by a si~tiplecontinuity argn~nentfrom our assumption that this is true for xo = 0 . ( I ) The theorem also holds for the elliptic operatom in two variable8 of odd order (6)

A'

~ $ 0 ,I))

zR.

the Diriohlet problettb

.

1 <j < 2 m Let, fzcrtL@r, u he a function such that u E L? (ASH)for some 1 and such that for all functions p E 0; (BR)the follozoing inequality holds :

q

>

where p' is some .lixed number

> 1 and

C i s a constant. 1

+1

Denote by p the exponent conjugate to p' : - - = 1 P P' u E H& ( S R ) . Moreover, jf 0 R' R and R, = ( R R')/2 then

+

< <

.

Then,

,

, , , , ,

where c, is a constant depending only on n m p l K R and R'. Proof: Assume first that j = 1 and that q > p , so that we also have u E L? (SR) In this case the theorem follows from Corollary 5.1 applied to u in SR, Next, let j = 1 but 1 < q < p . To prove the theorem in this case it will suffice to show that actually u E L?' ( f l R ) thus reducing the proof to the case just established. Now, let q' be the exponent conjugate to q . Since q'>p' it follows from (6.1) that we also have

.

.

,

for all functions cp E fl (BE). Hence, by the result just established ( p replaced by q) we conclude that u E H : ; ~ ( & ) .Invoking Sobolevls inequalities (Theorem 3.2) it follows that u E (SR) if either q 2 n or q n but 4, = q n / ( n - q) > p . On the other hand if q n and q, < p ~ o b o l e v ~ s inequalities give only that u E L$ (SR). In this case (noting that q, q) we repeat the same argument with q replaced by q, ; either arriving at the desired result zc E L ? (Ha) or at lea.st proving that u E L:' (SR)with q, q, Carrying on in this manner we obtain after a finite number of steps that u (SR) This yields the theorem for j = 1 To prove the theorem for j 2 we use induction - supposing the theorem is true for j - 1 ( 1 <j - 1 2 m) we shall prove it for j . We first observe that without loss of generality we may assume that A contains no terms of order (2 W E -f :

LP'

<

,

<

>

> .

ELF .

.

<

SHMUET. AGMON: The Lp Clppl~lIChto

,

(In the general case one can omit from A all terms of order < 2 9th -j the resulting operator will still satisfy an inequality of the form (6.1) in Put bai = Dia, and define BE, with C replaced by C 11 u

+

,

From the induction assumption it follows that u E HY?i,Lp(BE) and in particular that DiuE L? (SR) Set R, =(R' R,)/2. For V E Cr(SRo)we have :

+

.

Now, from (6.1) we get

Also, since

,

,

where DfDY= Da with (/3I =j - 1 ( ( 5 2 m - (j- 1) we find readily that

where p is a constant depending only on n and m . Combining (6.3), (6.4) and (6.5) we obtain the inequality

which holds for all functions

P) E

G(HR0) , i = 1 ,..., n , and

where

The inequality (6.6) shows that the derivatives Diu satisfy the conditions of the theorem in SRowith j replaced by j - 1 . Hence, using the induction assumption, we conclude that Din E (Sh) (and consequently

Di u € B ~ : , L(RE)) ~ . Also, denoting by n , m , p , l , ~ R, and R', we have:

ot constants ' which depend only on

the Dirichlet problem Thus (using Lemma 3.2) we infer that

U E H:;"?~ (SR) and

Finally, noting that by the induotion assumption we also have:

we derive from (6.8) and (6.8)' the desired inequality (6.2). This establishes the theorem. We now pass to the more refined result yielding regularity at the boundary. We shall first deal with functions defined in the hemisphere Z R , and with the regularity of such functions near the flat part of the boundary. THEOREM6.2. Let A ( x , B) be an elliptic operator of order 2nz dejhed i n j R , with coeficients satisfying Condition ( j ; K ] j being an integer such that 1 5j 2m I f n = 2 assunze i n addition that A' (ma D ) satisjies the < roots condition, for every $xed xo E Z R . Let, further, u be a functiolz belonging to L q ( . 2 for ~ ) sonbe q > l such that

,

< .

(zR) satisfying

for all functions p, E COO

D:P)= P)

on

,

the boundary conditions :

dixR for

k=O

= O i n some neighborhood of

,...,m - 1 , dz ZR,

>

where p' is Rome $xed number 1 and C is a constant. Denote by p the exponent conjugate to p'. Then, tb E Hj,I,p(ZR~)for every R' < R Moreover, setting R , = ( R R 1 ) / 2, we have :

+

.

,

,

tohere c, is a constant depending only on n m , p A , K , R and R ' . The proof of the theorem depends on Corollary 5.2 and the following lemma. L E ~ 6.1. A Let tc E L, ( Z R ) p I. Suppose that the distribtctiolt derivcrtives Di~ ) i ~ cfor i = 1 , , n - 1 are functions belonging to Lp( 2 ~Suppose, ). moreover, that there exist an integer 1 > 0 and a constant 0,such that

...

, > .

jor all .functions y E

. Then, the

Cr (ZR)

(distribution) deri-

vative D,&uis also a function belonging to Lp (ZRr)for every R'

< R , and

where c is a constant depending only on n , 1 , p , R and R'. The lemma in the special case p = 2 is due to Lions (see [19]). For general p (and also its analogue for Hiilder classes of functions) the lemma is given (essentially) in Agmon-Douglis-Nirenberg[3] where, however, instead of (6.11) it is assumed that

where fa are certain f~nctionsbelonging to Lp (ZR) and (6.13) is understood in the weak (distribution) sgnse. Clearly, (6.13) implies an inequality of the form (6.11). The converse implication is also true (see [191). For the sake of completeness me shall furnish in the sequel a variant proof of Lemma 6.1 which is not using the representation formula (6.13). We shall now give the Proof of Theorem 6.2. By the interior regularity result of Theorem 6.1 we know already that u E H:T~ (XR). In the following we shall assume that the operator A contains no terms of order (2m -j . This (as in the proof of Theorem 6.1) entails no loss of generality. Given 0 R' Et we set R, = (R' R)/2 and R, = (R' R,)/2. We also denote by ck constants which depended only on n t)t p l K , R and R'. To prove the theorem suppose first that j = 1 and that q > p so that, in particular, tc E Lp (ZR). Since in this case u satisfies the conditions of Lemma 5.2 in SR,,applying Corollary 5.2, we conclude that

+

< < ,

+

, , , ,

Diu€.Lp(.ZR0)

for

i=1

,...,12-1,

and that

To complete the proof in this case we need only to show that & u E L ~ ( ~ R ' ) and that

the Dirichlet problem

To this end write A in the form:

where a , $ is either a coefficient in A or zero. Let y E

( Z R J . We have :

Combining (6.16), (6.14) and (6.9) we infer

for all functions y E C$ ( Z & ) . Applying now Lemma 6.1 to the function a u in 2 % we conclude that D, (a u) E L~(ZR')and that

< <

<

Combining (6.18) and (6.1 4 ) (using 1-I I a I 1, I Di a I K) me conclude that Dn 21 E L p ( Z E r )and that (6.14)' holds. This yields the theorem in the case considered. Next, suppose that j = 1 but that 1 q p We shall reduce this case to the preceding one by showing that actually u E Lp (,ZR,) for every R' R. We proceed as in the proof of Theorem 6.1. By the above argument ( p replaced by q) me have 26 E H ~ , (L2 ~ 1 ) Hence, . using Sobolev7s inequalities we conclude that 2c E L p ( Z R l ) if q 2 n , ,or if q n but q, = q n/(n - q) 2 p On the other hand, if q n and q q, p we conclude that u E L,, ( Z B ) . Repeating in the last case the same argument with q replaced by q, etc., we conclude after a finite number of steps that u E Lp (zR') for every R' R This completes the proof of the theorem for j=1. Finally, for j 2 2 we use induction - assuming the theorem is true for j - 1 (1 j - 1 2 m ) me establish it for j . From the induction assumption we note that $4 € Hj-l,Lp( Z R ! )SO that in particular Di u € Lp ( 2 ~ 3 for every R' R

< < .

<

.

<

< .

<

< .

<

< <

<

Semu~r,AGMON: The Lp approacl to

Consider a derivative Di t6 with i f n . Let g, be a function belonging to Cm (ER) with support in such that

zRo,

1); g, = O

for

xn

=0

,

We now proceed to estimate (Di$ 4 , A g,)zR0 in exactly the same manner as in the proof of Theorem 6.1. Using the fact that Di 9 , i += n , satisfies (6.9)', wc need only to rewrite relations (6.3) to (6.6), replacing everywhere the sphere BRo by the hemisphere zR0. Rewriting the final inequality (6.6) we thus conclude that

,...,

satifor i = A - 1 and all functions 9 E Cm ( 3 ~ with ) support in fying the boundary conditions (6.19). IIere C, is the constant (6.6)'. Hehce the derivatives Di u ( i f: n) satisfy the condition^ of Theorem 6.2 in zK,with j replaced by j - 1. Applying the induction, setting .'R" = = (R' R,)/2 we infer that

+

,

From (6.21) and from the validity of the theorem for j - 1 it follows that all derivatives Da u such that 0 (I a I j a += ( 0 ... 0 j)belong to L P ( Z R , ) and satisfy

<,

, , ,

+

li Dau I I L ~ ~ Z ~I ~ ) cs (0 (1 u IIL~(XR~))

(6.22)

To complete the proof of the theorem me need only to show that D! u E Lp (zR') and satisfies (6.22). This me do again with the aid of Lemma 6.1. Write A in the form :

+p2 11-1

(6.23)

A =a

~

2

.i=l lal=2w-1

aa,iDi Da+

2

2m-jSlal52m-1

Let g, E CT ( Z R t r ) Using . integration by parts, we have :

a, Da.

the Diricldet problem

Consider a typical term in the first sum on the right of (6.24). Writing Da = D@DY with I ,8 I =j - 1 and I y I = 2m -j , we find readily that

,

Similarly, for a typical term in the last sum on the right of (6.24) we find (integrating I a I - (2m -j) times by parts) :

Combining (6.24), (6.24)', (6.24)" and (6.21) we thus get for all functions qJE C,m (2~") :

I 0 : - I (G) DmPin-j+l

(6.25)

9

)(

~

~

~

~

Applying, now, Lemma 6.1 to the function D:-' (k) in ZRtrwith 1 = -j 1 we conclude that Di (au) E Lp and that

= 2m

(xR')

+ ,

Finally, from (6.26) and (6.22) it follows that D: u E Lp (ZR') and that D: u satisfies an inequality of the form (6.22). This completes the proof of the theorem. We shall conclude this section presenting a Proof of Lemma 6.1. We first note that by an obvious approximation argument the inequality

(ZR),but also for all functions rp E 0'(ZR) with holds not only for pl E compact support in Moreover, we claim that (6.27) holds for all fnnctions p, E Cz( 3 ~ satisying ) the boundary conditions :

zR.

( D ; P = O on

1

di.ZR, j = 0

,...,1 ,

.

9 I0 in a neighborhood of 8, ZR

SHMUKL AGMON: The Lp approach to

Indeed, if g, is such a function and

E

> 0 , define:

g , (x) = 9 (x - E) for

1

,

5 6 ZR

for ~

g,, (x) =.O

8 2.

1

€ x <2e .

~

~

If s is sufficiently small then pl, will be a C1 function with compact support in &. Hence, by the previous remark, (6.27) holds for 9,.Letting E 0 we establish the same for g,. Now, define u as zero in xn 0 , I x 1 2 R Then, extend u into the half-space x, 0 putting

-

>

< ,

'+l

u(a,gn:,)=2 j=l

x' = (xi,

4~

.

( ,--7) for XI

xn

<0 ,

...,c ~ - ~where ) , the constants 4 are chosen so that 21+1

2

-

1

for k = - l , 0 , 1 ,

j-1

...,21-1.

Clearly u E Lp ((Sk) and

, ... ,

where here and in the following y, y, , denote constants depending only on I. Also, the distribution derivatives Di u for i n are functions belonging to Lp(SR) such that

Let, now, x be an arbitrary function of side SR . Write

,

+

( S R ) , extended as zero out-

Using (6.28) to transform the last integral in (6.30), we find that

fke Dirichlet problem

where

I t is readily seen from (6.31) and (6.28)' that x* is a Cm function with support in SR such that

D : % * = O for a n = O , Putting p, =

DL

j=O,

..., 2 1 .

x*, we write (6.30') in the form :

Since, now, p, E Cm (ER) and satisfies the boundary conditions (6.27)' it follows from the preceding that it satisfies (6.27). In terms of x (using (6.31), (6.32)) we have

I (*, D:'

(6.33) Next, for i f n

x)sRI 5 Y 3

ci 11 X ~ I ~ ~ - ~ , L ~ ~ ( Z R ) *

, we have :

from which, using (6.29)', we obtain that

Let A = D!' $

..: + D:' . Combining (6.33) annil (6.33)' we conclude that

.

for all functions x E (SR) Since A is elliptic, the inequality (6.34) allows us to apply Theorem 6.1 (Corollary 5.1) to u in SR. We conclude that u E H :;' P (SE)and that for every R' R (using (6.29)) I

<

, ,

where c is a constant clepending only on n I , p R and R'. This eseablishes the lemma.

SHMUICI. AGMON: The Lp approach to

7. The interior regularity in L, of weak solutions of ellipt,ic equntions and overdeternlirled systems. Let ( A i( x ,D)]:, be a system of N linear differential operators of respective order mi :

A i(x , D) = 2

a:

(x)Da

IalSmi

defined in a closed bounded domain G. We shall say that the system (Ai] is elliptic in 8 if there exists no real vector 6 =j= 0 and a point xE Cf such that

,

(7.2)

2 a: bl-mi

(4P = 0

for i = 1

,...,N

(p= g, ...Q).

If the leading coefficipts of Ai are continuous in there exists a constant 1 1 such that

>

G,

ellipticity implies that

for all real anit-vectors 5 and a E 8. We term such a constant an ellipticity constant of the system. For an overdetermined system of operators having the same order the above definition of ellipticity coincides with given by Schmartz 1201 (see also IIiirmander [15]).We point out, I~owever, that we are not imposing the restriction that the operators Ai be of the same order. In the following {~i):, will denote either an elliptic operator ( N = 1) or an elliptic overdetermined ,system ( N / 2) defined in 3 and given by (7.1). The formally adjoint A? of Ad is the operator A:(@,D) u = 2 (- l)laIDa (a: (x)U ) lallmi

.

I t is a differential operator in the ordinary sense if a,: E cia' (6). Clearly the system (AT) will also be elliptic. We shall consider a weak solution u of the adjoint system (7.4) in the sense that (7.5)

A?u=fi,

i=1,

...,N ,

the DiricSlet problem

.

for all functions g, E 0 : (G) Note that (7.5) has a sense when the coefficients of Ai are merely measurable bounded functions. The main interior Lp regnlarity result for such weak solutions is the following THEOREM7.1. Let u be a function belonging to (G) for sonze q 1. Suppose that u satis$es (7.5) where fi (i = 1 ,N ) are given functions belonging to L: (G), p 1 , and where ( ~ i ) : , ( N 2 1) is the elliptic systevn introduced above. Assume also that the coef$.cients of Ai satisfy Condition (1;61) i n Q , 1 beilzg some positive integer and put j = min (1, m, ... mw). Th'hen, u E H."~' (G) Moreover, i j Go, G, are any two subdomains Go, G, such that ALP

,...

>

-

Go c G,

LP

>

, ,

.

c , c G , then

,

where c is a constant depending only on n , max m i , p , N , K , the ellipticity constant 1 and the domains. Proof: Put nto = min m i , m = max m i , and let d be the distance bet-

xi

.

ween dGo and dG, Denote by the differential operator with coefficients complex conjugate to those of Ai. Given a point x0 E define :

6,

where A is the Laplacean. A& is a linear diiferential operator of order 2nt with coefficients satisfying Condition [ I ;co K ] in co being some consta,nt depending only on w nt and N . Also, A,o is elliptic at xOand conseqnently, by continuity, is elliptic in some neighborhood of xO. More precisely, since the coefficients of the leading part AAo possess first derivatives bounded hy c O K , it is readily seen that there exists a positive number Q < d , Q depencling only on n m N , K , 1 and d , such that

a,

, ,

, ,

<

for I x - xOI e and all real vectors t. Thus, denoting by S & , the sphere I x - SO I r , A& is elliptic in and 2A2 can serve as its ellipticity constant.

<

sd,,

(8) Condition 1 1 ; Kl for the coefficients of Ai is defined as in $ 5 (Def. 5.1) except that 2/11 .~houldbe replaced by the order mi of A i .

zp approuclh to

Sa~ulcr,Aciro~: The

(Sd,,). By (7.5) we have :

Xow, let g, E

which after summation yields : N

(7.7)

(21

,A@P)s@,,=i-12 (fi & 1

(gol

Dl

Am-"i

CT)#&,,

I t follows from (7.7) that

I

( u , Azo p)sd,,

1 I 0.2 llfi I L ~ ( S Z O , ) N

11 P Ila*n-%Jp~(s,n,e)

s-1

where 0 is a constant depending only on n , m , N , and K. The conclusion of the theorem follows now immediately from (7.8) and Theorem 6.1 applied to u in Sd,g (elliptic operator Ad), using a finite covering of 6, by spheres Sn$,e,2(xi E Go) The following is an easy consequence, and at the same time a generalization, of Theorem 7.1. THEOREX7.1'. Suppose that the conditions of Theorem 7.1 hold and tkat in, addition fi E lcX m N ) ( 6 ), ki 2 0 Set k = min ( 1 k, m, Then u E H;,?~ ( 6 ), and for any two subdomains G o , 6 , such tkat goc 6 , c c Cf, c 6 the following inequality holds :

.

.

, + , ... , +

+ ,

.

,

where c is a constant depekding only on lz , max (mi ki) N ,p K : l and the domaiws. Proof: The special case lc, = = kN = 0 is Theorem 7.1. In the general case put :

...

Ai,, for

IaI

= Hi

( 3 ,D ) = Ai ( X

, i = 1 , ... ,N .

,D ) Da

and fi,a = (- 1)4 Da fi

,

IntegraOing by parts we deduce from (7.5) that

...,

for c p € C ~ ( C f ) (, a I = h , i = l , N. The conclusion of the tlieorem follows now from (7.10) and Theorem 7.1 applied to the function u, elliptic system (Ai,,) and the corresponding system of functions ( f ~ , ~ ] .

the Dirichlet problem Suppose now that the conditions of Theorem 7.1' hold with ki=1 ) ~ nti and 1 = n~ It follows from the theorem that u E H$;~ (G). Using integration by parts it follows in a standard may from (7.5) that u is a strong solution (in H ~':) of the adjoint system (7.4). If, moreover, the conditions of Theorem 7.1' hold with k4 = m - ms j Ic = nz j where j n/p then it follows from Sobolev's inequalities that u E Cqn(G), fi E C(G) (after correction on a null set) and that u satisfies (7.4) in the classical sense. Finally, if the coefficients of the system and the fi are infinitely differentiable one obtains the well known result that u is also infinitely differentiable (for overdetermined elliptic systems see, for instance, Schwartz 1291). With the aid of Theorem 7.1 we establish now the following a priori estimates for a system of differential operators. THEOREM7.2. Let ( A ~ ] : , be a7z elliptic system of diflerential operators of respective order mi de$ned in 6. Set mo min mi, and suppose that the coeficients of Ai satisfy Condition (mi;K ) in G. Let Go be a subdomain such that goc G. Then, for all jbnctions u E CT (Go):

.

+,

+,

> ,

=

where c is a constant independent of u. Proof : Put Ai u =fi Then, for every function y

.

E

(8):

where ( A ? ) is the formally adjoint system. The inequality (7.11) follows now from (7.12) and from Theorem 7.1 applied to u in 8, system [A?) and l=j=mo. The estimate (7.11) for a single elliptio operator was established by various authors (see, for instance, Nirenberg [24]). For p = 2 and m, = 0 = m, = = n b estimate ~ follows from the more general Garding's inequality [14]. For general p the estimate (7.11) was (essentially) established in AgmonDouglis-Nirenberg [3 ; Th. 15.1"] by. a different method. In the special case of an elliptic system of operators having the same order the smoothness assumptions imposed on the coefficients of Ai in Theorem 7.2 could be relaxed considerably, namely, me have THEOREM7.2'. Let {Ai( x D)]::, be an elliptio system of operators in 6 ,

...

"

,

having the same order m. Suppose that the coef$cients of highest order terms in Ai are continuous, zclhereas the remaining coejicients are nteaszcmble aud

SEMUE~. AGMON : The Lp approach to

bounded in

g.

Then, for all functions u E C r ( G ) we /have :

where o is a constant independent of u. We sketch the proof. Using Lemma 3.4 we may assume without loss of generality that Ai ( x D) contains no terms of order nb Let z0 be an arbitrary point of G and put A: = Ai (xO D) By Theorem 7.2 the inequality (7.13) holds for the elliptic system with constant coefficients [ A : ) . Hence, there exists a constant co 0 such that for all u E (Q) we can write

,

< .

, .

>

Using the continuity of the coefficients of Ai it is readily seen that there exists a number Q 0 (independent of $0) such that if the support of u is contained in the sphere x - x0 Q then the last term on the right of 1 (7.14) is less than 1) u Ilm, L p(~) From this and (7.14) it follows that there 2 exists a number 8 0 such that (7.13) holds for all functions u E Cy ( G ) which in addition possess support of diameter 8 . Finally, one drops the restriction on the support of u in a standard way by using a suitable partition of unity and using once more Lemma 3.4.

>

I< ,

I

.

>

<

8. Regularity at the boundary.

We pass to the problem of regularity at the boundary in Lp of weak solutions of the Dirichlet problem. We consider an elliptic operator A of order 2m defined in G :

If n = 2 we assume in addition that A satisfies the roots condition in (i. e. for every SO E G the principal part A' (zOD) satisfies the condition on the roots introduced in § 4). We denote by C' [Da]lalsm-l) (nt 5 b ) the subclass of function v E 17' (6)satisfying the boundary conditions :

,

Dav=O

(a;

on a G for O < l a l 5 + ) t - l .

the Divichlet problem

We also recall that Hl,Lp(G;(Da]1611m-1) denotes t,he suubclass of functions

v E Ht,Lp (G) satisying (8.2) in the generalized (trace) sense (see $ 2). We now state the basic

THEOREM8.1, Let u be a function belonging to L, (Cf) for some q > 1 .

Suppose that for all functions v E GZrn(G ; {D")l,ls,n-l) the .following inequality hoWs :

I (U ,AV)GI 5 C 11 v 112m-j,~,,(~)

(8.3)

where A is the elliptic operatw (8.1), j is a positive integer 5 gm, p' > 1 and G a constant. Suppose also that the coeflicients 0.f A satisfy condition

where c, is a constant depending only on n , m , p ,E,1 (the ellipticity constant), and the domain. Proof: By an obvious covering argument it suffices to show that for every x0 E there exists a neighborhood Q' in the relative topology of G such that u E (Go), and such that 11 u I l j , L p( u j is majorized by the right side of (8.4) with a constant c, depending in addition on f P . For a point x0 in the interior this follows from Theorem 7.1, taking for Q0 a sufficiently small sphere with center at xO. Suppose that x0 E 8 G In this case there exists a sufficiently small neighborhood G of xO in 6 , and a measure preserving homeomorphism ( 9 ) of class CZm : I which takes 5 onto the hemisphere :1 / 1 )0 Let A be the transformed elliptic operator under the mapping and put Let, = u (x (G) and defined in further, be an arbitrary function belonging to C2n" (Zi; (Da)lalsm-l)and vanishing in some neighborhood of d,2, (the curved part of 82,). Put v (z)= Y(g(2))and extend v as zero in I t is readily seen that v E @In(G; (Da)l.;Bm-l). Using (8.3) we have :

.

zi

< , zw .

- -

(3

(z

a a.

One can take e mapping of the form : x, = 5, , ?-.,

(9)

...,

q).

SHMIJEI, AGMON: The Lp approach lo where co depends only on the mapping. Applying now Theorem 6.2 to the function ; in 2, we conclude that E; H j , L p for every r 1 and consequently that u E HjILp(go), @ being the image of Fr under the mapping. We also obtain by the same theorem the desired estimate. This establishes Theorem 8.1. From Theorem 8.1 one deduces easily the regularity up to the boundary of weak solutions of the Dirichlet problem :

(zr)

on

1Dau=O

aa,

<

o
THEOREM8.2. Let u be a function belonging to L, (0)for some q Suppose that u is a weak solution of (8.6) i n the sense tha,t

.

27n

>1 .

-

for all functions v E C ( G ; (Da)lals,n-l),where A is the elliptic operator (8.1) and f is a function belonging to L p ( G ) , p $ 1 . Suppose, moreover, that the coefJcients of A verify condition ( j ; K ) , 1 Ij 2 m , and that G is of class 02". Then, u E Hj,Lp( G ) and

Il u llj,~,ca)5 0( l l f

+ II

ll~,ce)

u

IIL,~~))

7

, , ,

where c is a constant depending only or2 n wz p K , l and the donhain. Proof: From (8.7) we obtain the inequality

for all functions v E C2*' (% ; (Da)la:sm-l),and the result follows by Theorem 8.1 A case of special interest is THEOREM8.2'. Ij the conditions of Theorem 8.2 hold with j = 2111. e. aaE Clal ( G )for I a > 0, a(",... , o ) being bounded) then u EHZw,~,(G; (D@Jl,ls,,) and satisjies (8.6) i n the strong L p sense. Proof: We know already that uE H27n,Lp(G)and consequently that A% =f in the strong Lp sense. To complete the proof we need only show that the trace of Uat&on the boundary (considered as an element of L, (da)) is zero for 0 5 1 a I Lwz - 1 For functions u which are sufficiently smooth this follows in a well known manner from (8.7). With some precautions the proof for functions of class H27n,Lp ( G ) is similar. For the sake of completeness we present a formal proof.

.

e.

I

.

the Dirichlet problem

I t suffices to show that given xOE dG there exists a neighborhood n of x0 on aG such that the traces y (Deu) ( I a 15 - 1) are zero when restricted to n. Since the last property remains invariant under a domain homeomorphism of class Omwe may assume without loss of generality that n is the n - 1 dimensional sphere x,, = 0 I x' I r (x' = (xi x,-,)). We may also assume that the cylinder : x' En 0 <xn 8 , for some 8 > 0 , belongs to a . Furthermore, noting that the trace y (D: u ) E H2m-l-j,Lp on n (this follows from the estimate (2.3)), and that

,

,

<

, ...,

<

for all derivatives D:, of order 1 a 1 2 2 m - 1 -j not involving xn we conclude that it will suffice to show that y ( D i u ) is a null-function on n ( j =0 9 n - 1). Let 0 <j j n z - 1 and assume that y (D: u) is a nullfunction on n for every l t s j - 1 (there is no assumption f o r j = 0). We shall show that y (DAu) is also a nullfunction on n and the result will follow by induction. Let p (8')E Cr(n) and let [(x*:,,) be a Cm function on x n ) 0 such that 5 = 0 for xn 2 8 5 (xn)= (- 0,,)2-l-j/(2n~ - 1 -j) ! for 0 xn 5 812 Put :

,

, ... ,

<

,

.

,

Since u E H2,m,Lp (G)we can integrate (u A w ) by ~ parts to obtain the usual Green's formula with boundary values taken in the generalized trace sense. A simple calculation shows that

,

where a is the coefficient of D? in A. Since A"u =f and (u A ~ G=)(f, ~to)@; we conclude from (8.11) that

(D: (a u))9)(x')ax' = 0 n

for all CJIE Cl;o (n). This implies that y (D: (a u)) and consequently y (D; u ) are null functions on n (since a 0), and completes the proof. Snppose that the function f in Theorem 8.2' belongs to L,(G). Then, by the theorem, u E H2nl,Lp (G) for every p so that, using Sobolev's inequa-

+

SHMUP~. AGMON : The Lp approach to

lities, u E CZm-I(G; (DaJlalSm-l) (40). If, moreover, f E Ci (G) and a, E Clal+l(G) then, by Theorem 7.1', zc also belongs to Prn(G)(ii). Thus, in this case u is an ordinary solution of (8.6). As a side application of Theorem 8.2' we mention THEOREX8.3. Let u be a function belonging to Qm(G)n Cm-I (G), suck that : A u = 0 in G , (8.12) Dau=O on a G , I u l ( m - 1 ,

I

where A is the elliptic operator (8.1). If G is of class Cw and the coefjcients a,, E Clal (G) then u E H2m,Lp (G) for every p. To prove Theorem 8.3 it suffices to show that

,

for all functions v E bzm(0; (Da)lallm-l), the result will then follow from Theorem 8.2'. This, however, follows from (8.12) by Green's formula applied to u and v in G (using a suitable approximation procedure). Theorem 8.3 is useful in connection with uniqueness theorems for the Dirichlet problem (for strongly elliptic equations) where it is necessary to assume in general that u E HmjL, (G). The theorem shows that this extra condition is really superfluous (12). The main applications of the regularity theorems will be given in Part 11. In conclusion we add only the following REMARK: Combining Theorem 8.2 with the a priori Lp estimates up to the boundary given in Agmon-Douglis-Nirenberg [3] one can show (with suitable regularity assuptions on the domain and the coefficients of A) that if in Theorem 8.2 j'E Hk,L# (G) then u E H2,*+k,Lp (G). Similarly, using the Schauder estimates in integral form of [3] one can show that i f f belongs to the Hiilder class Cktp(g) (k a non-negative integer and 0 ,u 1)'then u E C2m+kfr (C).

,

< <

<

More precisely, u belongs to the Hiilder class C2"-'+" (G) for every p 1 . We note that the same conclusion holds with the following weaker aasulnptions: f and a(o,,,, , are continnons fi~nctionssatisfying looslly a Hiilder condition aa E Clal (G) for 1 a 1 0 . For s proof see Agmon-Donglis-Nironberg[3; Appendix 51. (12) In this connection see.also Mirands [20] Lemma 11.1. (10)

('4)

>

,,

t l ~ eDiriehlet problem

BIBLIOGRAPHY I [ I ] AGMON.S., Multiple layer. potentials and fhe Uiriclflef problem for higher order elliptic equatioss i n the plane I, C e m n ~ .Pnre Appl. Math., Vol. 10, 1957, pp. 179-240. [ 2 ] AGMON,S., The coerciveness problem for integro-differential forms, J. d'Analyse Math., V o l . 6 , 1958, pp. 183-223. [ 3 ] AGMON,S., DOUGLIS,A. and NIRENBERG,L., Estimates near the bousdary for solutions of elliptic partial differential equations satisfying general boundary condition8 I , Comm. Pure Appl. Math., Vol. 12, No. 4, 1959. [ 4 ] AILONSZAJN, N., 011 Coercive integro-differe~ttial quadratic forms, Conference o f Partial Differential Equations, Univereity o f Kansae, Technical Report No. 14, pp. 94-106. [5] BABICH,V . M., 011 a problem of extension of functioas, Uspehi Mat. Nauk. N. S., Vol. 8, no. 2 (54), 1953, pp. 111-113 ( i n Russian). [6] BROWDERF. E., Strongly etliptic systems of differential eqtcations, Ann. Math. S t u d y 80. 33. Princeton University Press, 1954, pp. 15-51. [7] BROWDER, F. E., Oa the regularity properties of solutions of elliptic partial differential equations, Comm. Pnre Appl. Math., V o l . 9, 1956, pp. 351-361. 181 CALDERON A. P. and ZIGMUND,A., 011 aingalar integrals, Amer. J . Math., V o l . 78, 1956, pp. 289-309. [9] DENY J . and L I O N S J . L., Les espaoe de Beppo Leci, Ann. Inst. Fourier, V o l . 5, 195354, pp. 305-370. [TO] FICHERA,G., A l c n ~ irecenti sviluppi della teoria dei problemi a1 contorno per le equazioni alle derivate parziali lineari, A t t i Cong. Int. Eqnaz. Lin. der. parz. d i Trieste, 1954, pp. 174-227. [ l l ]FRIEDRICHS, K . O., The ideutity of weak and strong extensions of differential operators, Trans. Amer. Math. Soc., V o l . 55, KO. 1, 1944, pp. 132-151. [I21 FRIEDRICHS, K. O., On the differentiability of nolations of linear elliptic differential equations, Comm. Pnre Appl. Math.. Vol. 6, 1953, pp. 299-326. [13] GAGLIARDO,;E., Propriefd di alcune classi di futbzioni i n pi& variabili, Rio. d i Mat., Vol. 7 , 1958, pp. 102-137. [14] G ~ R D I N G , L., Dirichlet's problen~ for linear elliptic partial differeatial equations, Math. Scand., V o l . 1, 1953, pp. 55-72. [15] H ~ ~ R M A N D L., E I LDifferentiability , properties of solutioas of systems of differential eqtiations, Ark. for Mat. V o l . 3, 1958, pp. 527-535. [16] J O I I N ,F., Plane wave8 and spherical means applied to partial differeutial equations, Interscience. Ngw Y o r k , 1955. [17] KOSELEV, A. I., On bowudednesa i n Lp of derivatives of solutions of elliptic equations and systems, Doklady 'Akad. Nauk USSR, Vol. 116, 1957, pp. 542-544 ( i n Russian). [ I S ] LIONS,J . L., I'roblemBs aux limites eft tlteorie des dbtributioits, Acta Math., Vol. 94, 1955, p p 13-153. [19] MAGENES,E. and STAMPACCHIA, G., I problemi a1 contorno per le equazioui differenziali di tip0 ellittico, Ann. So. Norm. Sup. Pisa Vol. 12, 1958, pp. 247-358. 1201 MIRANDA,C., Teorema del massirno modulo e teorema di essistenza e di unicita per il problema di ~ i r i o h l e tredativo alle equazioni. ellittiche iit due variabili, Ann. Mat. Pnra Appl., Vol. 46, 1958, pp. 265-312. [21] M ~ R R E YC. , B. Functions of several variables and okolute continuity, Duke Math. J., Vol. 6, 1940, pp. 187-215.

Sanawr. AGMON: T h e Lp approach t o [22] MORREY, C. B., Second orcter elliptic systems of differential equations, Contributiontr t o t h e Theory o f Partial Differential Equations, Ann. Math. Studies No. 33 Princeton, 1954, pp. 101-159. [23] NIRENBERG,L., Remarks oir strongly elliptic partial differential equations, Comm. Pnre Appl. Math., V o l . 8, 1955, pp. 648-674. [24] NIRENBERG,L., Estimates and existence of solutions of elliptic equations, Comm. Pure Appl. Math., V o l . 9, 1956, pp. 509-530. [25] SCHECHTER, M., Integral iiiequalitiea for partial differential operators and fitnctioas satisfying general boundary conditions, Comm. Pnre Appl. Math., V o l . 12, 1959, pp. 37-66. [26] SCHECHTER,M., Solution of the Diriohkt problem for systems not necessarily strongly ed liptic, Comm. Pnre Appl. Math., V o l . 12, 1959, pp. 241-247. [27] SCHECHTER, M., General boundary value problems for elliptic partial differential equations, Bull. Amer Math. Soo., Vol. 65, 1959, pp. 70-72. [28] SCHWARTZ,L , ThQorie des distributions, Hermann, Paris, 1950-51. [29] SCHWARTZ,L., SU alcuni problemi della teoria delle equazioni differenziali lineari di tip0 ellittico, Rend. Sem. Mat. Fis. Milano, Vol. 27, 1958, pp. 211-249. [30] SOBOLEV S. L., On a theorem of functional analysis, Mat. Sbornik, N. S. 4, 1938, pp. 471-497 ( i n Russian). [31] STAMPACCHIA, G., S p a una classe di funzioni in n variabili, Ric. d i Mat., Vol. 1, 1951, pp. 27-54. [32] V I S H I K ,M. I., On strongly elliptic systems of differential equations, Mat. Sbornik, Vol. 29, 1951, pp. 617-676 ( i n Russian). T h e Hebrew University Jerusalem, Israel

E~t~.ttI,t,o dxgli Aicrtrcli dslltr ficitoltr Norabtcle Jktpsriore di Pisci

Serirr 111. Vol. XIV. FRRO.I (1960)

MULTIPLE INTEGRAL YROBLEMH IN THE OALOULUS OB' VARIATIONS AND RELATED TOPICS by CHARLESB. MORREY,JR.(Berkeley) (")

Introduction. I n this series of lectures, I shall present e greatly simplified acconl~t of some of the researcl~ concerning multiple integral problems in the calculus of variations which has been reported in detail in the papers [391, [401, [41], [42], [44], [46], and [47]. I shall speak only of problems in no~rparametric form and shall therefore not describe the excellent resnlt coilcerning double integrals in parametric form obtained almost concurrently by Sigalov, Danskin, and Cesari 1621, [9], [5]) nor the work of L. C . Yonug and others on generalized surfaces. Some of my results have been exteiidrtl iu various ways by Oillquiui [6], De Giorgi [lo], Fichera [17], NobBliag [51], Sigalov [58], [59], [60j, [61], Silova [63], and Stampacchia [67], [68], [69], [70]. However, it is hoped that the results presented here will serve as a11 introduction to the subject. The first part of this research reported in these lectures is an extension of Tonelli's work on single and double integral problems in which he em. ployed the so-called direct methods of the calculus of variations ([71] through [78]). His work was stimulated, no doubt, by the succes of Eilbert, Lebesgue [31] and others in the rigorous establishment of Dirichlet7s priaciple in certain important cases. The pri~icipleidea of these direct methods is to establish the existence of a function z minimizing an integral by showing (i) that the integral I(#)is lower 'semicontinuous with respect to some

(*) Presented at the international oonfere~loeorganized by C.1.M.E ill Pisa, September 1-10-1958.

CHARTXSB. MORRIEY JR.: Multiple htegrnj

kind of convergence, (ii) that I (x) L d for the x considered and (iii) that there is a <( minimizing sequence >> z, such that I&,)d and z, z, in. the sense required. In the case of single integral problems, where

-

-

Tonelli (see, for instance [76]) mas ableto carry through this program for the case that only absolutely collti~~uous functions are admitted, the convergence is uuiform, and (esseatially) f ( B z ,p) is convex in p (ifj'(m ,z p ) lf,,(p) where f,(p)/ 1 p I o , it is seen from the proof of Theorem 2.4 below, that the functions in any miuiluizing sequence would be uniforlnly absolutely conti~~uons so that a subsequence wonld converge ui~iformlyto ; ~ uabsolutely conti~luousf u ~ l c t i ozo ~ ~which would thus lninitnize I(z)). Tonelli was also able t o carry through the eutire program for certain double integral problems usil~gfunctions absolutely continuous in his sense (ACT) and uniform convergence [77], [78]. However, iu general ha had to assume that the integraud f (x y z p q) satisfied a condition like

-

,

,

,, , ,

I f f satisfies this condition, Tonelli showed that the functions in any mii~imiziug sequence are equicontinuous, and uniformly bounded on interior domains a t least (see Lemma 4.1) and so a subsequence converges 1111iformly on such domains to a function still in his class. H e was also able to haildle the case mbere (0.3)

f ( x , y , ~ , P , q ) ~ 9 ~ ~ ( 1 " q ~ )k- if

f(",Y ,z,O,0)=0,

for instance by sllowing that any ~ninimizingsequence can be replaced by one in which each z,, is monotone in the .sense of Lebesgne (see (311 and [37], for instance) and hence equicontinuous on interior domains, etc. However, Tonelli mas not able to get & general theorem to cover the case where f satisfies (0.2) only with 1 < a < 2 Moreover, if one coiisiders problems involving v > 2 independent variables, one soon finds that one would have to require a to be > r in (0.2) in order to ensure that the functions in any ~ninimizingsequence would be equicontinuous on interior domains. To see this, one needs only to notice that the functions

.

a,re limits of ACT functions in which I, B(0,l)

'1

I

dx and I V .1

I* dx for k < v/(h + 1)

B(0,l)

respectively, w e unifor~nlybounded (see below for notation). In order to carry throngh the prograin, for tliese more general proble~ns, thon, the writer found it expedie~itto allow fuuctions which are still more ge~ienllthan Tonelli7s ACT functions. One obtaii~sthese more geueral fullctions by merely replaciug the requlre~rieut of v-dimensional continuity iu Tonelli's defiuition by snmniability, but retaining Tonelli7s requirements of tlbsolnte continuity along liues parallel to the axes, summable partial derivatives, etc. But then, two such fuuctions may differ on a set of iueas~lre zero in such a way that their partial derivatives also differ only on a set of measure zero. II, is clear that such functions should be identified and this in doue in forming the <spaces %A3 discussed in Chapter I. These inore general functions have bee11 defined in various mays and studied by various authors in various connections. Beppo Levi [32] was probably the first to use functions of this type in the special case that the fu~ictionand its first derivatives are in J2; ally function equivalent to such a function has been called strongly digerentiable by Friedrichs and these functions and those of correspouding type involvi~lghigher derivatives have been used extensively in the study of partial differential equations (see [2], PI, ill], [Is], POI, 1211, 1241, 1281, 1301, [421, [451, [46l, [471, [BOI, [671, [61], [66]), G. C. Evans also made use a t an early date [14], [16], [16] of essentially these same functious in connection with his work on poteutial theory. J. W. Calkin lieeded them in order to apply Hilbert space theory to the study of bouildary value problems for elliptic partial differential equations and collaborated with the autlior ill setting down a uulnber of useful theorems about these functions (see [4] and [40]). The fuuctions have been studied in more detail since the war by some of the writers inentioned above and by Aronszajn and Smitli who showed that any function in the space H,,,(see Professor Niremberg's lectures) can be represented as a Riesz potential of order m [I]. The writer is sure that many others have also discussed these functions and certainly does not claim that the bibliograplly is complete. I n Chapter I, the writer presents some of the known results concerning these more general functions. In Chapter 11, these are applied to obtain theorems concerning the lower-semicontinnity and existence of minima,

CHAET,ICS3. Moartlc~ JR. : Multiple integral

of multiple integrals of the form

where the function f is assumed to be continuous in ( x , z , p ) for all ( 8 ,z , p ) a'nd convex in p = (p:) for each ( x ,2 ) . I n Chapter 111, the most general type of function f ( x , z , p ) for which the integral I ( z 0)in (0.5) is lowersemicontinuous is discussed. I n Chapter IV, the writer discusses his rescllts concerning the differentiability of the solutions of minimum problems. I n Chapter V, the writer discusses the recent application by Eells and himself of a variational method in the theory of harmonic integrals, We consistently use the notations of (0.5). I1 p, is a vector, Ip, 1 denotes the square root 0.f the sum of the squares of the comphents. Our functions are all real-valued uilless otherwise noted. If x is a vector or tensor z, z a p , etc., will denote the partial clerivatives dz/dxu d 2 z/dxa a d , etc., or their corresponding gei~erltlizedderivatives. Repeated indices are summed unless otherwise noted. If G is a domain dG denotes its bo~indaryand G = G U dG. B ( x o R) denotes the solid sphere with center at xo and r:bdius R ; we eometimes a.bbreviate B ( 0 , R) to B R [a, b] denotes the closed cell aa l xu 5 ba. A11 ir~tegralare Lebesgue integrals. I t is someti~nesdesirable to consider the behavior of a function (or vector) z ( x ) with respect to a pa,rticular vari;ible xa ; when this is done, we write x = (la ,$A) and z (z)= z ( x u ,2;) where xi stauds the remaining variables ; sometimes (v - 1) dimensional integlals

,

,

,

,

,

8:)

dxi

a,:

appear in which case they have their obvious significance. We say that a (vector) fnnctioi~z (8) satisfies a uniform Lipsohitz condition on a set 8 if snd only if there is a constant M such that

I z (xi)- z (x2)I IM I xi - x2 I for x,

and x2 on 8 ;

x is said to satisfy a uniform Holder condition on S with exponent p , 0 < p < 1, if and only if there is an M such that

I z ( x , ) - z ( ~ , ) I < X ~ 1 x -x,Ir ,

for xi and

l,

on 8 .

proble~nsin the calculus etc.

A (vector) function z is of class Cn on a domain 12 if and only if z and its partial derivatives of order I. s are coutinuons on G ; x is said to be of class 04+r or Cn on CS if and only if z is of class 0" on 0 and it and all of its partial derivatives of order I. s satisfy uniform Hiilder conditions with expoueut p , 0 < p < 1, on 0 ; the second notation 0; is used when p = 1 (see Chapter V).

Function of class qA, %i ,93;(11 1) and functions which are ACT. We begin with the defllritio~isof these classes: DEFINITION: A f~lnctioli z (m) (x = (xi, xv)) is of class oe a do~saijc U if and only if z is of class on G and there are f ~ l ~ ~ c t i op n* s, u =I , Y of class &A on G with the followillg property ; if R ie any cell with closure iu G , there is a seqtle~rceZ,~Rof fouctions of class 0' on H U dR such that s,,- 2 and zn,,-pa strongly in 2' on R. DEFINITION:A fiinctioll 21 is of Class %?i OU C f i f alld only if (i) a is of class &A 011 G ; (ii) if [ a , b] is m y closed cell ia (f tlleli z is AO (tibsolutely continaous) it1 xa on [au, b y for i~lniosta11 xi on [ a : , b:] a = 1 v; (iii) the partial derivatives a,, which exist' almost every-where and are measurable on accoant of (ii), are of cltiss &A on G . DEFIN~TION: A fuuction z i s class 9;011 B if and only if z is of class on G and is continuous there. D E F ~ N I T I ~A N : functiotl z is abaolt4tely co~ctinuous in the sellse oj' Toxelli (ACT) 011 G if and ouly if z is of class 93; ilncl is oontinuous on G . DEFINITION: Suppose z is of class &, on 0. We define its 1~ average function on the set Gh by

...,

..., ,

,

,

,

, ...,

(v

ith

zh (x)= ( 2 1 ~ ) -z~( E ) 115

,

X-h

+

Gk being the set of a11 x in G such . tlrt~tthe cell [s - lb, x h l c G. LEXMA1.1 : I s z ,is of class ZAole a d o l ~ t a C G trnd Zh i s its 1b.avevage as h 0 09, each closed cell [ a , b] fi~lcction defined on Bh thetb zh z i a 4 G awd zh i s contittuous O I L ah. Proof: That ah is continaoas follows from tlre tlbsolute continuity of the Lebesgue integral. Next, it is well known that zh (x) x (m) as 1b 0 for almost all x. Finally, choose IL, > 0 so tlltit [a - ho b k,] c G , keep 0 < k < lc, , and let y (g) be ii function 0 as g 0 such that 11 z lie 5 y [m (e)] for e c [a - lb, b Lo] where

,

-

-.

-

, + ,

-

-

, +

-

problems ithe calculus eto. Then the le~~lrna follows, since since

where e(5) is the set obtained by translating e along the vector 6. TEEOREN1.1 : I f z is of class q Aon G , the functions pa are cvnipael?y detevrbined up to null functions. I f zh is the k average of a attd pah is th(rt of pa then zh is of class G' on C f h and

,

, +

,

,

Proof: Let [ a , b] c U choose W, so [a- h, b JL,~c G imd keep 0 < h < h,. Approximate to z and pa by z, and x,,,, ia 2 2 on [a - h, b W,]. Then for e ~ c l lh , me see t l ~ n t aflh;,= (z,,,)~ and me lnay obtain (1.4) by letting a - oo on [ a , b]. The first stictement is now obvioas. DEFINITION : If z is of class %IA on a domain G , me define its generalizet derivative D, z ( x ) as the Lebesgue derivative a t x of the set fuuction

, +

/pa ( x ) h.

THPOREX 1.2 : I f z Xi8 of C ~ Q S S % oni Cf, zh is its h-average ficnctiox, and pah is that of its pavtial deritative &/axa, then zh is of class Or and (1.2) holds. Moreover z is ?f class %Aand its corvesposding partial awd generalized derivatiyes coincide a111tost evevywhere. Pvoof: Let [ a , b] c G , choose W, so [a - ho b k,] c U and keep 0 < h < h,. If x: is not in a set of measure 0 OII [a: - I&,,b: h] then azlaxa = p a ia summable in ma over [aa- hq ba W.,] alrd

,+

, +

,

+ ,

By integrating (1.3), we see that i t holds for all x: on [a;, b:] and all a ; , x; on [ a a ,ba] if z and pa are replaced by their h-averages. hen (1.2) aud ,the laat statement follow.

C H A R L ~ SB. Moarte~J a . : Multiple integ~al

THEOREM1.3: (a) I f zi atcd zz trre eqccivalent and one i s o f class on 6 , then both are and their ge~re~alizeddersiuatives coincide. (b) I f z, and s, tare o f class 9& O M a doa~aitt G and z,,, (x) = (3) a111lost everywhere o n 6 , t h e i ~ z, and z, dbfer by a consttrnt and a null .function. These are easily proved using the haverage filuctions. THEOREM1.4 : (?) A n y function z o f class on G i s equivalent to a function a, o f class 93i on Q b) z i s A C T ota G if and only if z i s of class 93; there. Proof: To prove (a), let R = [ a , b] be any ratiolinl cell in G and approxi~nateto z there by functions z, of class 0' on [a, bl. A subsequence, still called z,, converges to z almost everywhere and is such that

.

(ma n-w

,$3- z,, (aa, 4)IQxa = 0

E

, ,

for it11 x; uot in a set ZRa of ( V - 1)-dimensioual measure zero, u = 1 ... v. Prom (1.4), me see that the x,, (xu ,xL) are equicontinuos in ma and converge uuiformly on [ a a , b*] to a function XOR (xu .c;) which is A C in za if 3: is not in ZRa, u = 1, Y. Obviously zoR= x almost everywhere on R. Since the uuiou of the ZRafor u fixed ancl K running over all r;ttional dells is still of measure zero; we see that the $ 0 join ~ up to form a function 2, of cPass 93 ou G. To prove ( b ) , we note first that if z is ACT on 6 , it is of class 93; on G. Conversely, if x is of class qy,we may repeat the first part of the proof taking z, as the lh, -:tversge of z and conclude that we may take z o ~ always = z since then z,, couverges u~~ifornily tJo z on R. T l ~ efollowing theorems are easily proved by approximations : THEOREN 1.5 : l'he space 93, o f eqllic~tleaceclasses o f functiotis of class % i s (G Banack space we define the ~aovqnby

... ,

I f rZ = 2 ,

,

i s a real Hilbert space (f we de$tte

problems in the calcultts stc. THEOREM1.6 : I f t2 E aild i s qf c l a ~ sC' alrd sfitisfles n tc!~ifOrv)t Lipschitz condition on the bouirded donrain G , then i~u 6 % ow Q and the getteradized derivatives (hu),, all exist at nny point x, where all the zc,,(x,) exist. onto ff I)EFINITION : A trtlusformation T :x = x(y) from a domain which is of olass a ' is said to be regular if and olily if T is 1- 1 and T and its inverse are of class C' and satisfy uniform Lipschitz condition (1 x (Y,) - 8 ( ~ 8 I)S31 I Y, - Ye I, etc.1. THEOREN1.7 : I f u i s of class (93;)olt the botcnded demain ff, x=x(y) i s a regular trnnsfort~aatiolzof olass C' from the bounded dontaii~2 ofit0 Cf alzd (y) = u [x (y)], then i s of class (%:) on Cf Moreover, if ~ , = x ( y ~ ) and all the gelieralized derivatives u,, (x,) exist, then all the generalized derivatives G,, (yo) exist atzd

.

-.

P r o o f : That % is of class %I(%;') and that we may choose the right sides of (1.5) as the ct derivative functiolrs B of the definition is easily proved by npproximatilrg u on interior domains by fuuctions of class U'. S i ~ ~ cregular e families of sets correspond under regular transformations, the last statement follows easily. REMARKS : I t is prqved in [40] arltl [47], for instance, that if zc is of clttss %A on a, it is equivalent to a fu~iction (namely the Lebcsgue de-

FP

./

rivetive of udx) which is of class

x

% and i is such that any trausfor~rias

in

Theorem 1.7 retsins this property. But the last statement of Theorems 1.7 does not hold for the partial derivatives since this wonld imply that z had a total differentiill almost everywhere central-y to an example of Sake [55]. I t is clear how to define the gerleralized derivative in a given direction and that (Theorem 1.7) if 2111 the u,,(x,) exist, then a all the geiaevalixed directional derivatives exist at x, a ~ are ~ d given by tlreir usual formulas there. It is now easy to prove Rademacher's famous theorem [52] that a Lipschitz function has a total differential almost everywhere : For lrsing the result just nle~rtio~~ed together wit11 Theorem 1.2 we see that if z is Lipschitz and x, is not in a set of measure zero, then the partial and generalized derivatives all exist at xo and the ordinary directional derivatives in a denumerable everywhere dense set of directions (independelrt of 3,) all exist and are given by their usual formulas; at ally such poilrt z is seen to have a totd differential. Thus in Theorem 1.6, h may be Lipschitz illid in Theorem 1.7, the transformation and its inverse 1na.y be Lipschitz; in this case (1.6) holds whenever a11 the geaavalized derivatives involved exist.

CHAKLESB. MORRDYJR.: Nultiple integral

THEOREM1.8 : The ~itostgeneral lkear fictcctioaal on the space the form

% is oj

4

where the A, (a 2 0 )E with 1-1 f p-1 = 1 i f 1> 1 or are bounded and ~~ceasurable ole G if 1 = 1. Proof: Let An be the space of all vectors y = ( y o , ,y,) wit,h components in and

...

,...,

From Theorem 1.5 it follows tl~tbtthe subspace of all vectors (z,z , ~ z,,) for mlrich z ou f f is a closed linear manifold M i a Bn . Hence if P (2,zgl z,,,) =f ( z ) the11 P can be extended to the whole space B to have same norm aef. Then P is given by (1.6). From Theorem 1.8 we immediately obtain: THEOREM1.9 : (rc) A Necessary a ~ szcflcient ~ d co~cditiojcthat z,, converges on G. wecr,kly to z (z, 7 8 ) ia % on G is that z, 7 a a~rdthe z,,,, 7 z,, C (b) I f zl,7 . 2 in %Aos a, theit z, 7 x in %AO D nny subdoweni~. or f f (bounded), x = x ( y )is a regttlar trassforma(c) I f z, 7 z i n w tion of class 0' from owto ff, ; ,( y )= X , [x (y)] and z ( y )= z [x (y)],then F, 7 Fin on a. (d) I f a,,7 z i n qnon f f (bounded) and h is Lipschitz on f f , thew hz, 7 ha igc %Aon ff. DEFINITION:A fu~ictionz is of class 5Yjio on G (bounded) if and only if it is of class C)& there and there eviste a seqnence { t i n ] , each of class C' and vanishing on and near tlre bolllldary dG such tllat 2,- z (strong convergence) in % on ($. The subspace %A,lof %Ais defined correspondingly. If z i1.11d2"E %Aon G, we say that11 z = z* on dG itc the %a sense if and only if z - z* E CX3no on G* The follawiug is imn~ediate: is a closed liltear ntalcifold ,is %A; THEOREM1.10 : The sttbspace qAo ifz,, 7 z be %Aoic G a d each z, E%,,then z E %lo. I f z E 9 1 0 and z, (8)= z(x) for o on G and z, (x) = 0 otherwise, then z, E %'A~ on auy I) 3 G and z,,,(x)= 0 for alntost all x not i ~ tG. THEOREM1 .ll (PoincarB7s inequality) : Suppose z E %AO on GC B(xO R). Then

,...,

,

%

,

pvoblen~sin the cnlczrlus etc. R o o f : I t is s~ifficicient to prove this far x of cl:bss C' and va~iishing on dB (so,R) with G = B (so,22). Taking spherical coordinatei3 (v,p) with r = I x-xol and p E 2 = d B ( 0 , I ) , me obhin

where

21 (1.

,p) = 2

(3). Tl~iis

from whiclr the result follows. on 6, A c Q z' E %A 0 t h /1 trnd coinciTHEOREM1.12 : Alppose z E des with z on dA ill, the % se~ue.Then the ftcvtction Z such that Z(X)= zY(:z) on A altd Z ( x )= 2; ( x ) on Q - A i~ qf cltrss o s a irnd z,, ( x )= x:(x) nlnzost euaryzohere O N A alcd Z,, ( x )= z,, (3) trlntost ccverywheve on G - A Proqf: For define 2, ( x )= a" (x)- x ( x ) on A i ~ l d0 elsewhere. Theu Z ( x )= z (x) Z, (a) on G a11(1the res~iltfollows from Theorem 1.10. LEMNA1.2 : Suppose z E 93~on tlrt: cell [a - h, , b h,]. Then

,

.

+

+

6-l-h

zh(x)-~(x)~*dx~Ci(v,~).~~L.~ll~z(y)lAdy, a-h

where Ci depetids olaly ola the argztsaents indicated. Proof: Since we may approxin~ateto z stroi~glyin %Aon [a- h, b h] by f~inctiollsof class C' on that closed cell, it is siifficient to prove the lemma for sucl~functions. Then if 3 E [a, b] and 1p ( 5h , we see that x and

+

CHARLESB. MOKEEY JR.: Multiple integral

104

a + [ are in [ a - h , h + b ]

so that

Then

+ [) - z (x)]df IA dx

(z)- z ( 8 ) I A dx =

[z (x -h

from which the result follows. TFIEOREN1.13 : If z, 7 zO i n qoA O N the bou~tdeddoniail~G, thea 2,-z, iu JAon G, ,I> 1. I f (z,,)is IG seque~toe in %oA tuitlb (1 zn 11 unijor11tly bot~nded, a stcbseq~beltoeoonuergec strongly ill 21to some futtotion z. Proof: The first statement follows from the second. For, let (z,) be any snbsequence of (z,,]. A subseqrrence { z g ) converges strongly in 2 2 to some function z which must be (equivalent to) 2,. Hence the whole sequence z,, zo ill gA. To prove the second sttttement, suppose (f c [ a , b] and extend each z, 1 with uniformly to be 0 outside 0 ; tl~ell each z,,E C M o ~ 011 [a - 1 a 1 bounded 91ilorm. For each h with 0 < h < 1, me see that the Znh are uniformly bounded and equicol~ti~~uous on [ a , b]. So there is a subsequence, called {z,), such that zph converges uliiformly to some function zh for each h of a sequence 0. Froln lemma 1.2, it is easy to see first that the limiting q form a Oauchy sequeuce in @. A having some limit z and then that z, 2 strongly in JL. I n order to treat variational problems with fixed boundary values, one can, of course, practically always reduce the problem to one where the given boundary values are zero. Althoiigh one call formul:~te theorems about variational problems having variable boundary values on the boundary of, an arbitrary bounded dolnltin (see Chapter 11), suc,h problems become more

-

, +

-

-

problem in the calculus etc. meaningful if we restrict ourselves to domilins U which aro bounded and of class Cr where bouudary values can be defiued in a more definite way as we now do. DEFINITION:A bounded domain G is of class 0' if and only if each point xo of the boundary dQ is interior to a neighborhood N ( x O )ou G a O which is the image, under a regular transformation x = x ( y ) of class C', of the half-cube Q+ : I ma ( < 1 for u < v and 0 5 xv < 1 , where x ( 0 )= xo and dff fI N ( x o ) is the image of the part of Q+ where xv = 0. Such a neighborhood N(s,,) is called a boundary seighborkood. DEFINITION : Suppose G is a domain. A finite sequence (h, h N )of fuoctions is said to be a partition of ullity of class Or on G U dG if i~nd only if each hi is of class C' on G'UdG, O l h i ( x ) l l on O U d a for each i,and

u

,... ,

N

2 h i ( x ) f i for x on O U d O .

i=l

Tlie support of 1bi is the closure of the set of all x on O U dG for which hi (x) 0 LEMMA1.3 : If G is botrltdrd dol~~tti~b of clnss Cr, there is n pnttitio~of tiftity (li, liN) of cltrss Cr on U U dG sucW flint the suppovt of each hi is either interior to a call ilc G or is inta~iov to a botcfbdary ~teigliborlboodof G U dG. Proof. With each interior point P of Q we defirle Rp a8 the largest hypercube Ixa-$;I <<Wp ill G and define r p as the I~ypercnbeI xa-x;l < < 1ip/2. With each P on dG', t1ssociat;e a bou~idaryrreighborhood R p =N ( P ) which is the image under zp of &+ as ia the definition; we define r p as the part of R p corresponding under zp to tbe part of Q+ for which 1xa1<1/2, cc = 1 , v. There are a finite number v, r~ of the r p which cover Cf U dCS. Clearly each corresponding Ri is the image under a regular transformation y of class 0' of either the unit cube Q or the half-cube Q+ where ri corresponds under zi to the part where 1 xu 1 < 112. Now, let 9 ( 8 ) be a fixed fullction of class C" for all s with ~ ( s=) 1 for I s 1 2112 , q (s)= 0 for 1 s 12 314 and 0 5 9) ( 6 ) < 1 otherwise. For each i,define ki(x) on Ri as tlie image under q of the f~illctioll p(yi)...9(yv) and define $ ( x ) = 0 elsewhere on G U d(f. The11 the support of ki is interior to Ri lci (x)= 1 for 8 on ri and each ki is of class C' on G U dQ We then define

> .

,...,

...,

,...,

,

,

hi (x)= ki ($1

,

,

.

i

,

hi+l(x) = ki+~(8).n [I- kj (41 i = 1 9 I--1

9

If

- 1*

Then me see by induction that

hi (x)

+ ... + hi (a)= 1 - n [1- lc, (x)] Z

1-1

...

so that the sequence [hi, , lbN] satisfies the desired conditions. THEOREM1.14 : Snypose G is bottnded ccntl of clnss Cr and z E 0th G. T h f n (i) tt~ereis a seque9,ce (z,] of ft~nctioj~sof closs 0' on Cf U aG tuhicl~ converges stvoftgly i n 931 to z 0th G; (ii) thew is a boundary vtrlne futcrtiolb p, itr &AO N dG (with respect to Ibypernreu) to which every sepzcajcca [x,,) in (i) colwerges sttvongly i n 2 1 on a@; (iii) {/' T : ,r: = x (y) is n regultrv tvtr~~qfir~,lntion of oltrss Or of GU d z

a

w

oltto G U dG, 2 (y)= s [a (y)],a~td pl (y) = y [x (y)] , then $ is tlte bouttdnry value ~urtctior~ fo9.G or&ai4 ; (iiii) i j p (a) = 0 for (t1~110st all x 07) dG, tlcett z E 9 1 , O N ti. proof: Let (W, h N ) be a partition of uuity on U U dG of the type descyibed in Leiuma 1.3. Clearly each functiou his € 9~ oil O ant1 OII Ri a,uil the thansforul wi(y) ur~dsr ziE 9 1 OII either Q or Q + ; in the former c;lse wi vanishes or1 aud near d Q I L I I ~in the latter, wi va~~ishes near a&+ n 8 s . IUthe latter case, wi is eqaivillent to a ftl~lctiorrw~ which is AC ill y h for aln~ost all y i , u = 1 .. Y 011 BUY cell where I& 5 yv < 1 (since .mi = 0 near yv = l), where h > 0 . But sillce 1 ~ iE ~&,A~,we see that tea is A 0 in yv for 0 1y v S 1 for almost all y:. If we extel~clwa to the whole of Q by setting ula (yv y:) = wio (- yv ,y;) for - 1 5 yv I 0 N

,... ,

, ,

+

,

,

we see t,lrat wio E %'; ou Q and vanisl~esuesr a&. Clearly we may approximate each wi or woi on Q stro~lglyin gLby functioirs w,,~of clttss 0' on Q lc11i.I vanishing near dQ. If we defiue z,,i on Bi as the transform of q , i under zi and then defiue z,, = z,l $ s , , ~ , we see that x,, has the desireii properties. To prove (ii) we choose, in all cases, tot, eqt~ivaleut to rri aud 9; on Q. Then, since w* is A 0 in xv, me see that

...,+

1 I'

,

Accordingly, we see that wio (yv y:) coliverges stroi~gly in &Ain y: to z,, of class Of, aird we let w,i be the wio (0 ):y as yv O+. If z, z in tralrsfor~n of biz,, under zi, thee we see tha.t (1.7) llolds uniformly. Now let [p) be s n y sabsequence of (n). There is a subsequeuce (q] of [ p ) stich that (for each i) wqi(yv,yi) converges strongly in f i wit11 respect to y: on [- 1, 1 ] for almost J1 yv, 0 < @ L 1 But, on account of the nnifonnity in (1.7), this convergence is nnifonn for all yY,0 < yY51. Herrce tlre whole sequence wio (0 y:) converges strongly to wio ( 0 , y:) in &A. , . , (iii) is now evident. To prove (iiii), [x,,) be of elass G" and converge on G. The11 yn and each hi;,, Converges strolrgly to 0 strongly to z in on dG. If we define tire woi as above, tlrei~zoOi( 0 , yb) = 0 for alnrost all y:, on [+ 1 11 if Ri is a bo~undi~ry ~~eighborlrood. If we extend such woi to Q by woi (9') Y:) = - woi (- yv, y:) yv< 0 we see that z n ~is of clnss %A 011 Q 2nd that i t and its h-nve~agefunctio~rs,for sufficierrtly stnall h vanish near d Q and along yv = 0. By ~nodifyillgthe a,ver:lge furrction sligl~tly for each h in a sequence 0 we rrray c o n s t r ~ ~ csequences t w,,i tending strotlgly in to ?aoi= 0 such thot each W,i = 0 near yv = 0 a.s liear dQ for those i for whiclr Ri is n boundary neighborhood. The desired B , ea.ch of class C' alrd vanishing near dG call be constructed as above. THEOREM1.15 : If G is boutrded a d of class C' ccnd if z, 7 z irb ott G, thew z, x in &AON @ alzd y,, y ilz 2 1 on d G . 11' 11 x, 11 ia t~tbi.fOrf~tly

-

-

,

.

,

,

,

-

,

-

-

j Ip

bouttded in %A,a d t h e set fu~tctiofts

r

z,

1 dm

a1.e tcnifort~tly absolzitely

contitzzbous if ;f = 1, there is a subsequeitce (z,) which cosvrrges znea,kly in to sonte z on G . Proof: Let (h, hN] wlli, wi, and w ~ ihave ineaniligs as in the proof of Theorem 1.14 and let w,,i be of class %; on Q (or Q+) and be equivalent to w,,i and extend each wmoito Q as before. Then (1.7) holds u~iiforrnly (in case IZ = 1 this is true on account of the ulriforln absolute continuity in that case) and w,i- wi in &Aon Q for each i. The argument in the proof of (ii) in Theorem 1.14 c ~ be n repeated to obti~inthe desired results. The last statement follows easily. I n the next section, we shall have occasion to discuss vector functions of class DEFINITION: A veator fu~iction x = (xi, zN) is of class %'A if and only if each of its compolients i s ; in this case

,... , ,

... ,

CHARI~IES B. NORRICY J R . : Multiple integrnt

It is clear that all the theorems and lemmas of this section except Theorem 1.11 aud letr~ma1.2 generalize immediately to vector functions. Those two can be generalized with the help of the folloming well known lemma : LEMMA1.4 : Stcppose f , f, are sum7)table over the set S toitlh respect to the ntaesure p. Phew is also arid

- /1

,...,

Proof: For the left side of (1.8) equals

I n addition, me need the follomi~~gspecial case of Rellich's tl~eorem 1531 :

side

THEOREM1.16 : If the vector z i s of clnx8 %Aon the hypercube R of 1b t r d zR i8 its nvernge over R, tihen

where C8 depetlds o7d?y ow the nrguntents isdictrted. Prooj': I t is s~ifficie~lt to prove this for vectors of class 0' milere R : xa I < k = k / 2 . The11 we have I .-

~robtentain tAs calculwe etc.

Settil~gqa = xa

+ t (p- ma)= (1 - t ).wa 3- t $ , the last integral bscoines

I

wlrere R(q,t)is the intersection of R wfth the hypercube ~ @ - q " / ( l - t))
1 q - x 1 5 ~ 1 1 .%tlb . The result follows since

st

,

[ R(k t)]5 hy and is 5 (2th)" for t 5 1/2.

CHABLICSB. MORRICYJR. : Multiple integral

Lower-semicontinuity and existence theorems for a class of multiple integral problems.

In this chapter, we consider variational problems for integrals of the form (0,6) in which f ( 3 , z p) is continuons in ( 8 , z p ) for ill ( x , z p) wnd is convex in p for each ( x , z) (of. [42], Chapter 111). DEFINITIONS : A set X in a linear space is said to be convex if and only if the segment P, P, belongs to S whenever the points Pi and P2 do. A fullction p, (8) (6= (ti, EP)) is said to be convex on the convex set 8 in the &space if a n 4 only if

,

,

,

...,

whenever 5, and & € 8 . The followil~g.tlleorems concernil~g convex functions are well known a,nd Itre stated without proof: LEMMA2.1 : 8uppose g,(E) is convex on the open conves set 8 with / g, (5)I 5 M there. Then p, satisjies

any compact subset of 8 at a distance 2 6 from d B . LEMMA2.2 : Suppose 9~ and each p7, are convex on the open convex set B and suppose p, (E) p, (6)for each 5 on 8. Then the convergence is uniform ow any compact subset of 8. LEKMA2.3 : A necessary and sufjicient condition that g, be convex oa the open conves set S is tlbut for each in B there exists a linear function (I,. EP b such that oa

-

+

(2.1)

p,

(6= a, P + b , p, (t) a, 5 P + b

for all

If g, is of class C' otr X, this condition is equivalent to

E E8.

problems in the cnlculus

eta.

I f rp is of class C" on 8 , this condition is epuivale~ttto

-

Y,afi(E) va 17'

>0

for all ? on B and all 17. DEFINITION: A linear function a, 6, b which satisfies (2.1) for some 2 is said to be supporting to p, a t ?. LEMMA2.4 : Jlzppose p, is oonves for all 6 and satis$es

+

, ,.. ,

Then q~ takes 0th its ntiaimzcm. Also, if ai ap are any numbers, there is a unique b such that a, EP b is supporting to p, for some E . I f y, is convex and satisjies (2.2), if jf (6) 2 p, (6) for each 5 , and if ap 5p c is snpporting to y , then c x b . LEMMA2.5 : Buppose that p, and p, are everywhere convex and satisfy (2.2) and suppose that p,, (t) p, (6) for each t #upposs a, , a, are any nun8bers and b, and b are chosen so that up 5 P b, and a, 6p b are s ~ p portitag to rp,, and p,, respeotively. Then b, b Likewise, i f a, -ap for each p and b, and b are chosen so that a, 6 P + b, and ap EP b are all supporting to f, the* b, b . I n order to consider variational problems on arbitrary bounded domains, it is convenient to introduce the following type of weaker than weak convergence in 33, on such a domain. DEFINITION:We say that Zn z0 in %, on the bounded domain G if aud only if z, and zo all E %, on B , zn 7 z, in %, on each cell interior to C: and each z , , 7 zo,, in 2,on the whole of G THEOREM2.1 : If Q is bouaded and of class C' or i f all the z, E %,, on 2, in 33, on Q , then z, zo in qi on 6. Q and if z, Proof: The second case can be reduced to the first by extending each z, to be zero outside G and choosing a domain r of class 0' such that T DG . Thus we suppose CS of class a'. If we use the notation in the proof of Theorem 1.14, we see that (1.7) holde uniformly for the wn, so that an arguu~elit siluilar to tilose in the proofs of Theorems 1.14 and 1.15 and 1.13 shows that w,,i converge strongly in k?, on Q or Qf to something for each i . Thus z,, converges strongly in Ji on Q to something which must be z, REMARK:If Q is not of class 0' and the 2, are not all in qio on 8 , the11 an example in [41] shows that z, + 2, in 9, on Q w i t h o ~ ~ t

+

+

-

,...

.

-

.

+

+

+

-

.

T)

.

CHARI.ES B. MORRICYJR. : Multiple integrat

the

q, norms of the zn being uniformly bounded. If for some 1>1,

11 -1 F h , dz

G

[i;:,.Py

( C bounded)

0

are uniformly bounded, the11 a subsequence { p ) of (s)exists such that the z,,, 7 something iu 2, on the whole of C . THEOREM2.2 : fJ2cppo~ethat f ( p ) is dejlzed of all p = ( p i )(i = 1 , N a =1 v) and f is convex. I f z, + zo oil C and

...,

, ... ,

,

thelz I ( x , G) and I ( z , , Q) are each j ~ i eor

,

+ oo and ,

I (z, C)5 lim'inf I (8% 0). n-to

Pvoof: Since f is convex, there are constants a; such that

for all . p Hence

with

R similar inequality for I ( z , ) . Thus the first statement follows. If D C C we see as above that

,

by virtue of the uniform sbsolute continuity of the set functions

,

,

(z)d z e

.

Clearly also I (z D)- I (z C ) as D runs through an expanding sequence of domains exhausting C . Thus i t is sufficient to prove the lower semicontinuity for G a hypercube of side h , say. To do this, we define a sequence of summable fulictions y q ( x ) as follows: k'or each p divide C into 2'9 hypercubes of aide h 2-9. On each

problems i# the calculus stc. of these hypercubes R , define

where pia is t h average ~ of zfa over R and the a; (pi -pia) is supporting to f a t p~ larly from 2 , . Then i t follows that

+

f (pH)

id;

.

( R , q) are chosen so that W e define the pl,, simi-

(almost everywhere). On the other hand, suppose all the genertdized derivatives exist a t some xo which is not OIL d R for any hypercube R as above for any q . Let R denote the llypercube containing xo Then as q oo pia nfa (8,) so that yq (8,) f [Vz (a,)] since the a; remain bounded (Lemma 2.5). Hence

IVq

~ ( z 8) , =~ i m 9. 6 '

(2.3)

-

.

-

pa-

h.

,

Moreover, for each fixed q p , --pR ~ f r o ~ nthe weak convergence so dx = 2 f (pR)111 (R) = liln 2 f (pllR)111 (R) = R

G

=liln

11-m

I

R

yfsq(x)d x ~ l i l n i n f I ( z , , , O ) .

n-w

n-w

G

The result follows from (2.3) and (2.4). LEMMA2.6 : Suppose j'( a , z p ) is dejiaed aid, satisfies n unifovolal Liyschitz coitdition with consttrv~tK for all ( a , z p ) szbppose f (x z p) i s convex iia p for each (a, z) aed suppose f (x x p ) 2 f,(p) f b r n l l (m x p ) , where f, (p) is convex. Then, if x, 7 zo in on CS

,

, ,

Ti

,

, ,

, ,

, ,

,

I(#,, C)
Proof: A s in the proof of Theorem 2.2, i t is sufticient to prove this for a hypercube D of side d interior to C3 Then xn x0 in 2Aon D Prom the Lipschitz condition, f(x,z,p)~f(O,O,O)+K.IxI+K.lzI+K.Ip1 so that I ( # , D) and I @ , , D) are finite,

.

,

-

.

CHABLESI).

~ ~ O B B EJB. Y

: Hultiple integral

For ei~chp , divide D into hypercubes R of side 2-q. d . Then, using Theorem 1.16, it follows that

< IL . 2-q [2-1 -

1I

vlIz. dw+

,lim cq = 0

Vz(x)Iax]

9-8,

D

and a similar inequslity holds for each z, with account of the weak cotivergence. Also

i~idependelitof n on

E~

T l ~ elemma follows easily from Theore111 2.2 and the inequalities above. THEOREM2.3 : #uppose f ( x z p) is deJined ~ c s d oontinz~oiisfov all ( x z p ) , is convex in p fov each ( x z ) l a d f ( x z p) j; ( p ) for all ( x a , p) where fo ( p ) is convex and fo ( p ) /l p 1 oo as p oo . Thejb I ( # ,a) is lower semioo~~ti~iuous with respect to the coavergence 7 Proof: I n order to prove this, i t is sufficient to show that f ( x , z , p ) is the lilxiit of a noii-decreasing sequence f, ( a , a , p) each of which has the properties required in Leirima 2.6. I n order to do this, let b ( x 2 ;a ) (a = (a;))be chosen so that the function ( x z ;p ; a) = a,;pi b ( a , e ; a) is the unique supporting plane (in p) to f determined by a . By Lenimas 2.4 and 2.5 b ( x z ; a) is continuous in ( x z ; a) and b ( x z ; a) 2 bo ( a ) , the corresponding function for fo For each a , choose a non-decreasing sequence b, ( x z ; a) of functions, each 2 bo (a)- 1 each satisfyng a nni'form Lipschitz condition for all ( x z)., which converges to b ( x z ; a ) . We then define rp, ( x z ;p ; a) = a;pk b, ( x z ; a) and we see that y, is a non decreasing sequence tending to y for each a , each y,, satisfying a uniform Lipschitz condition everywhere. For each n , we define j ~ ( x , ~ , ~ ) = m a x ~ , ( x , z , for p , aall) a for which all the a; are rational numbers haviug nunierator and denominator

, ,

,,

,

, ,

,

-.

-+

+

,

,

,

,

,

.

,

+

,

,

,

,

,

problems in the calc~clusetc. both 5 11. Theu it is clear that tho f,, are non-decreasiag and each satisfies a uniform Lipschits conditio~r.Now, let (x, a,, p,) end 8 > 0 be given. Usiug Lemma 2.5 and the continuity of b , we see that there is a ratiol~ala, sue11 that v(z0 ,Po; a) >f(xo,Zo ,PO) -42. Clearly v n ( ~ , g0 o , Po ; > 93(xo,zo ,Po; a ) - el2 for all sufficiently large n so that f , (xo zo ,po) f ( x o ,po). We now tarn to existence theorems on arbitrary (lomeins. We begin with the following tlreorem (cf. [48] and [40],theorem 8.8 and [41]: THEOREM 2.4 : Suppose *fo( p ) is coltsex in p attd fo ( p ) /1 p oo as p m. Thett there is (6 function 9 (e) 0 as Q 0 wkicll 8epends o d y o s f and M such that i f I ( z , 13)s M , then

,

,

-

,

-

-

4

-

,

I -+

I'roof: For each integer r > 1 , let E, be tlre set of x in G where r - 1 5 1 V 2 ( x )I< r and V z ( x ) exists and let

where Z is the set of measure 0 where V z (a) does uot exist. Clearly Lo = G and if r~ 1 a1111 $ 8 O - &,, theu I V #(%)I
From this we see that

-

-

.

and both 0 as r oo So, let e be any subset of @. Let. r be the smallest integer such t h t ~ tdllr 5 m (e). Then

and a, satisfies the conditions.

,

THEOREM2.5: Suppose f (x z ,p) strtisjies the hypotheses of Theore9,i 2.3 atid G is a botc~rdeddoi~tain.Suppose that r Xis a family oj"fic~ccfioirszX i w %, which is cosrpact with respect to the convergence i i n 9, on G . S q p o r e F is the f(1mily of all a i s %, which coincide on dG itr the 98,sense with some zX ill r* and suppose F contains some for which I(< , G ) < CC. Theti 1 ( 2 , G) takes o~cits nrini~)btcqibilt F. I'voof: Let [z,,) be a minimizing sequence (i. e. I (a,, greatest lower bound for z in P ) ; we lnay assume that I (z,, G)5 M = I G). Suppose a,, = a: on Q where z: E r X . A subsequence 8:: i zo+ in 93,on G and

+

6

,

-

, a)

Fi ,

./I

Vz, I dz are ul~iformlyAG'; the

same is true of the set fur~ctions V (aq - 5:)

I dx . Since G is booiided and

a$ ErX. By Theorem 2.4, the set fuiictions

I I

;a,

eacl~zp - z i = 0 on we see with tile aid of Theorem 1.13 that n sub. sequence z, - z: 7 some w, in %, oiid G ; ~ u dw, = 0 on aG. Accordiugly 2,. 7 z, = zf w, in %, on G tulcl a, E F . the theorem follows from the lower-semicontinuity of I ( z , G). Somewhat more meal~ingful boitlldary value problems can be studied if we require G to be of class 0' a t least. We need the following preliminary lemma : LEMNA2.7 : Suppose G is bozcnded a d of class 0' and F is a ftr~ibily of futtctions of on such that

+

a

Suppose that P satis$es one of the following additional coibditions : ( i ) there is a niiatber P and aft open subset a of G such that

j

Z I U X L Pfor "11 a E B ; o r

(ii) there is a nunrber P and an opeir set o of dG such that

Then the

nomts of the

n!

in

I? are usifor1nly bounded.

problelnr itc the calctclcs etc. I'roof: W e may cover G IJdG with a finite nnalber of hypercubes or boundary neigl~borhoodsR, , ,R Q ;let yi map 0 or Qf 'onto Ri es in the 1)roof of Le~nma1.3. W e IrjiLy ctssnlne that one of the Ri c z i l l case (i) or thilt Rin dG C o ill case (ii). In case (ii), we see using equation (1.7) with y; = 0 that case (i) 11ol(ls wit11 z = Ri rt,ttd P rel)lacerl by Pi; here we lrlive assumed that wio is equivaleat to the trnllsforln under yi of tlle restriction of z to Ri. Now, let R j n Ri be an open set t i j . For a given x , let wj, be of class % ofiQ or Qf 8nd be equivalent to the transform] u1id6r pj of the restriction of z to R j . Thus there is a cell Rjo= [a, b] in & or &+ such that case (i) holds with x replaced by zcjo ailcl I' by Pi,, (ilrdepender~tlyof x in P). By using ttn equi~tionlike (1.7), we see in tun1 that case (i) llolds with Rjo replaced R j l , lrjz, Rjv= Q or Qf wit11 I' replaced by Pjl, Ij, = Pj' where R j 2 i s t l ~ e c e l l - 1 ~ x ~ ~ 1 , - 1 ~ x 2 ~ 1 , a a ~ x ~ ~ b ~ fv o, r u = 3 , etc. Thus case (i) holds with z replaced by Ri and P by P i . Since any Rk can be joined to the first Ri by it scquelrce R,, each two adjilcent nlelnbers of which have a n open set in colnmon, the lemma follows. W e can now prove our second priucipal existence theorem : THEOREN 2.6 : 8uppose the domtrin G and the family P satisfy the conditions of Lee~wttc 2.7 for some 1> 1 (and hettce for I = 1 and xtbppose P colttainx some vector for which I (F, G) is finite and stbppore P is closed with respect to zceclk coneergejbce in %, . Suppose that f ( x , x , p ) strtisfies the conditiotcs of Theorem 2.3 Then I (2, G) talces o,t its ~ ~ b i t ~ i miunmP. Proof: Let ( x, ) be a miuimiaiug sequence for which I (x,, , G ) 5 I @, 8).

...

...,

...,

I

Then the set functions

...,

dx are uniformly absolut,ely continuous on ac-

B,,,

e

count of Theorem 2.4. Combining this with Theorenl 1.15; we see that a subsequence [x,) can be selected which collverges weakly on G in 93,to some x, in %,. Since P is closed with respect to weilk convergelrce in 2, z, E E. The result follows from B e lower semicontinuity of I ( a , G). THEOREM2.7 : Suppose G is of class C', f (x , z ,p) satisfies the hypotheses of Theorem 2.3, altd i s a closed favtily of functions y i n Pi on dG such that case (ii) of Lemma 2.7 hold^. Stippose P is the fatitlily of all functions x i n 9, on G, each of which has boundary valties i n and suppose E coetains tc ficnctiolc iuch that I G) is finite. l'hen I (x G) takes ow its minimum in P. Proof: For the snbfamily %' of 8 in P for which T ( z , G)< I @ ,G) satisfies the conditions of Theorem 2.6, on account of Theorelus 2.1, 2.3; and 1.15.

,

r

(z,

r

,

C H A R L ~ B. S MOBRICYJE. : dlultiple integral

EXAMPLE : As a'11 exampl'e of the use of Theorem 2.7, consider the problem of findiug the surface z =. z (3)(2 = (xi, 2" zx3), x = (xi, s2))of least area of type of a disc bounded by a simple closed C consisting of a fixed arc C; whic11 has only its end points on a surface S and a veriable arc C2 011 S. Usillg tl~eoremsabout conformal mapping t l ~ i sprobleme can be reduced to that of miuimisiug the Dirichlet integral

among all vectors z of class %{ on G , where G is the unit circular disc, such that the restrictions of x to dG carry the upper semicircle of dG in a 1 - 1 continuous way onto the fixed arc Ci with (0,l) corresponding to some fixed point on C, and carry the lower part of aG in a 1 - 1 continuous way onto the variable arc C2. In order to apply Theorem 2.7, we let consist of al! stroug limits in gt 011 dG of the restrictions of such z to dff . Ally vector g, in is equivalent along the upper part of dG to a vector which carries that pmt of aG in a < mopotone B way onto 0, in which arcs of C, may correspond to poil~tson dff; far a~mbstall x on the lower part of dG , y (x)E S at ally rt~te.Siuce any minimizing vector zo certainly mi~~imizes I ( z , G) tl~mougall z ill q2wllich coincide with x, on dG in the %2 seuse, we see that xo is harmonic (see Professor Nirenberg7s lectures). By arguments like those in [7] and [43], we conclude that xo is colltiuoous on the upper half of dG and yields a conformal map of G onto the surface represented by 2,. However, an exaulple of Oouraut [8] (p. 220, 221), shows that zo need not be continuous along the lower hdf of dG and t h t t,he limiting e curve B C2 need not be an arc even if the surface S is regular and of class Om; Lewy [33] has shown that if 8 is analytic, the curve 0, is analytic.

r

r

pvoblesis in the enlcztlus etc.

Quasi-convexity and lower-semicontinuity. I n the preceding chal,ter, we proved theorems concerning the lowersemico~itinuityof multiple integrals I ( z , O) in cases where the integral function f ( x , z , p ) is oontinuo~~s and coavex in p for each ( x , z). This restriction on f was a ~iaturalexteuaion to the case of several unknown functions of the ordinary requirement when N = 1 that the variatiot~al problem be regular or a t least that ~ a d a h a r d ' s condition

.hapg(x,*" P) 1, A!

0

,,

for all (a z p , I )

be satisfied, f being assumeti of class 0". But (3.1.) holds if aad only i f f is convex in ( p i , ... ,p,) for each (xi, xv 2). The condition (3.1) is arrived at as follows : Slippose a function z, (x) of class C' mit~imizes1(z, O) among rdl fut~ctio~ls of z of class C' which have the same boundary values and which are near z, in the s e ~ ~ that se the maximum of I 2 (x)- zo (x) 1 V z (x) - V zo (x) 1 5 d for some 8 > 0. Tl~euit call be sl~owr~ that (3.1) holds for x OII Q ,z = zo (a), aud p =VzO(x). However, if this procedure is applied i n the case where N > 1 we obtai~i only the condition

... , ,

++

,

,...

for all (x ,x , p ) (along the solution x = z, (x) , etc.) and all (1, ,A,) and (fi ,tN) (see Theorem 3.3 below). This does not imply that f ( 8 ,x , p ) is convex in p. Moreover, it is known that integrals 1( z , O) which arise in parametric problems are lower semi-continuous wit11 respect to ul~iformconvergence; for the case of the parametric problem for surfaces in 3-space (v = 2 N = 3), these integrands have the form

, ...

,

where J 1 - p , 2 ~ ~ - J~2 ~= p~ ~~p ~~ - p ~ p J ~3 )= P I 1P2-p;pi

, ,

and P is convex in (J, J, J,), but uot in the six p i .

CHARLICS. B. MOREICYJR. : Multiple ijctegml

It turns out to be rather easy to derive (see also [44]) ti eertain neces. sary slid sufficient couditio~~ ou f as a function of p for the lower semicontinnity of 1(z 6 ) wit11 respect to a certain type of convergence. This qnestion wys aonsideretl for Y = N = 1 by Tonelli ([72],[73],[74],[75))and by Cesari and others for the pt~rametriccase. We begin by deriving this condition slid then discuss the relati011 of that coridition to the condition (3.2). I n order not to get involved with the behavior of f wt infinity we shall use the following convergelice which obviously implies weak convergence in each k'%' but doea not ~~ecessarily imply stroug couvergence in any DEFINITION: We SHY that z,, z on 6 z,, (3)converges ur~iformlyto z ( x ) on 6 aud z and z,, each satisfy a 1111iforrnLipschitz condition on 6 which ie illdependent of n THEOREM3.1 : Slippose I ( z ,6 ) is 1ower.sen~icoittinzcous with respect to this type of convevgettce at nny z oil any 6 and f i s coittinuous. l'lien,

,

+

s:

--

.

for any cotistant (xO, g o , po), aity bouttried dotttain Q, and aity Lipschitz vec-' tor 5 tohich vanisher oa 8 6 . Proof: Let x, be any poilit, R be the cell x; xa x; h 8, be any vector of class C' ou R U aR Q be the cell 0 5 sa5 1 , and 5 be any vector which satisfies a uniform Lipschitz condition over the whole space and is periodic of period 1 in each sa. For each n , define 5,(x) on R by

<

,

+ ,

+

Then the [i teud to zero in onr sense. Then, for each w , I ( z 0 C,, R) can be written as ib snm of iotegriils over the sub-hypercubes of R of side 9'-I h. If r is one these the iutegral over it is

where w : x;
z,

x;=x;+

k a ~ r l hO, < b 5 n - 1

(m)= zD(8)+ 5, ( x ), xu = x; + n-I

h

, 0 5 a (1 .

prob~ems in, the cnicuius etc.

Thus we see that

.

By letting 2, and p, be arbitrary col~stantverctors, setting 2, (x)= z, +poa ( R )= hV a11d lettillg h -- 0 , we obtain (3.3) for (xu- x;) dividing by G = Q ;~pd( periodic of period 1 in each ma. But if G is any bounded domain and 5 vanishes on d a we may choose a hypercube Q' containil~g G and extend 5 (8) to be zero in Qt- U . The11 e simple change of variable obtains the result in general. DEFINIT~ON : If f is contiliuous in ( x z p) for all ( x z p) and satisfies (3.3) for all (x, z, ,po), me say that f is quasi-eonvex in p ; if f depends only on p and satisfies (3.3), we say simply that f is quasi-convex. We now prove that the col~dition(3.3) is sufficient for lower-semicontinuity. LEMMA3.1 : Suppose R is the hypevcube I xa - a; 15 h ,f ( p ) is quasi-convex, suppose p, is any constant tensor u,%d szvppose 5% 0 in o w se~teeor R. Then

.

,

,

,,

,,

,

-

I

+ V 5,, (a)]dx >f (PO)

lim inf f [po n-w

9"

(R)

Proof: Suppose the (,, satisfy a uniform Lipschitz condition with constant Y on R. We may assume that 1 (, ( x )1 < Ylc, k where each kn<1/2 and liln kn = 0 . For each n we begin by de611ing 9, (x)= (n ( z ) on d R and 7, (x)= 0 for I xu - x; 15 (1 - kN)h ; we then extend each q, to the whole of R to satisfy a Lipschitz condition with constant 5 M . Then m 0 , c n - q,, 0 (, (x)- q,, (3)= 0 on d B , and (x) 0 for each x interior to R. Henee

,

-

-

d,,

,

-

The result follows easily froq the quasi-convexity off. THEOREM3.2 : Xuppose f ( x ,z ,p ) is quasi-coavex C p, G is a bounded dowain, and z , zo on G. Then

-

,

1 (z, G) lim inf I ( z , , n+w

a).

CHARI.LESB. MORRLCYJK. : Multiple integrut

,

,

,v

Proof: Since all the arguments [x xn (x) 2, (x)] and [x, zo (x) vao(x)] remain in a bounded part 9 of (x, y ,p)-space and since Cf is the union of C3e, disjoint hypercubes, it is sufficient to prove this for the case of a h y yercube R of side h. Since f is uniformly continuous on 9,there is a function (Q)= 0 such that E (Q)with lim e-0

If

(x', 2; p') -f (%", z", p")2 & (Q) if

1%'

- x'l

l 2 + 12' - 2" 1-

p'

-p"12 5

For each k, divide R up into 2~~hypercubes Rkj of side 2-" h. Define the functious x,$(x), xf (x) p$ (x) on IZ to be equal on each Rki to the averages over ltw of 8 ,z, (m), aud..yo(m) respectively, thud define

,

t n (4= xll ($1

- 20 (4.

= f 1% 7

($1 9 210 ($1

Then

where

Bnk

(3.5) Ck = f [x

$0

+

an

($11

- f :;1'

,20 ,PO(x)] -f [ d (4, (2)

22

($1 1 ZX

, PZ (')

+

(x) pk* (41; (a,,(m) = p C,,

,

(4)

We see that

,

+

+ +

Lk PHI,,where these are the inteand I@,, R) - 1(2, R) = J , grals of A- HItk aktand D a n respectively. Now, let e > 0. We first choose a fixed k sucl~tliat KHkaud Lk are both < &/2.From (3.4), (3.81, (3.6), and Lemma 3.1, we see that

,

,

,

problems in the calculus etc.

,

,

since xf ( x ) z$ ( x ) and p; (x) are each constant on each R M .Thus

liminf[I(z,,R)-I(B,,,R)]>-&. n-m

Some of the theory of Chapter 2 can be carried over for the more general functiolls f ( x z , p ) which are quasi.convex in p but more has to be asaulned about how f behaves as p - oo These theorems are not of great interest and they can be found in 1441. We now investigate the concept of quasi.convexity in more detail. LEMMA3.2 [79], [46] : Suppose a?@ave constaltts and

,

.

Ik

for all

1'

in

qZo on. domain

Q, the%

a;/ 1, Ap f j

tk

0 for all I and

5.

...

Proof: Let A' be a unit vector with A:= 1, and choose 1') , I' so (Ai, ... lv)form a nornial orthogonal set. Suppose xo E Ct and let yy = 1;. . (LF- 5).ghoose A, and 1Z > 0 so that the set of all x for whiclr 1 yll
,

where and y ~ and , y ( 1 yi 1 ) = 0 otherwise. Then it is easy to see that lim (2A)-l. I(5, , Q )= f,,-lRY a$A, Ig

E k / v (V

k-0

- 1) 0

which proves the lelnmaWe now prove the theorem mentioned in the introduction to this chapter. THEOREM3.3 : Buppose f ( x , 2 , p) is of class a'' for all ( x , z , p) near the locus S of all points [ x , z , (x), z0 (x)]f o ~o in Q and suppose zo ( x ) is of class C' on Q U dG artd alinin$izes T ( z ,a) anaosg all Lipschitz z which coincide with zo on 8 Q aud ave such that z ( x )-zo( x )1 ( p z ( x )- z0 (2) 5 8 for some 6 > 0. Then (3.2) holds for all ( x x ,p) on & Proof: For, let 5 be any Lipschitz functioli vanishing vanishing on and near dB. hen 2, 15 is sufficiently near so for a11 sufficiently mall

v

I

+

,

+

v

I

Csaar.vs B. MORRICYJR. : Nultiple integral

1 So if ~ ( 1=) l ( z o

+ A[), we must have

By selecti~~g any point xo in G and proceeding as in the proof of Lemma 3.2 alrd the11 dividing by Rv/v (v - I)], but letting R and h both 0 so that k : B 0 , we obtain (3.2) at [x,,, 2 (xo), p (xO)], Using the result of Le~nlr~tb3.2 and the method of proof of Theorem 3.3, we conclnde that i f f ( p ) is quasi-convex and of class Cn, then (3.2) holds wit11 s and z omitted. This ~.esnltand the analogy wit11 convex functions suggest the followi~~g theorem whichwe IIOW prove. THEOREM3.4 : I f f ( p ) is quasi coltvex, the14 f ( p i 1, lj) is convex i n 1 for each p and 8 and convex i n 6 for enoh p and 1. Proof: I f f is quasi-convex, it is easy to see that its twiceiterated haverage f11nctio11jhh ie a180 quasi-convex and is of class C" as well. Then any linerir function furnishes an absoliite minimum to Ihh ( 2 , a) among all Lipschitz functions with the same boundary values. Accordingly, by Tl~eorem 3.3 we see that fhh satisfies (3.2). But then fhh has the convexity properties stated in the theorem, Since .rhh collverges uniformly to f on any bouuded part of space, the theorem follows. DEFINITION: A function f ( p ) which satisfies the conditions in Theorem 3.4 is said to be weakly qnasi-convex. REMARK : The principal problem, so far unsolved, is whether or not every weakly ,quasi-convex function is quasi-convex. THEOREM3.5 : If f ( p ) is weakly quasi-convex , it satisfies a unifornb Lipschitz condition on a bounded pcwt of space. I f p is given, there are coltstants A; such that

-

-

[r,

+

f ( p h + l a € j ) >-f ( p ! ) + A;IZa[j for all 1 , t . I f f is also of class O', then Aa==f,i(p). I f f is also of class CN then (3.2) 3 a holds. I f f is continuozcs and if, for eachp, constants A4 exist such that (3.8) holds, then f is weakly quasi-convex. Proof: I f f is weakly qua,si-convex, it is convex in eachpi separately. Hence, if If ( p ) l i ; M on some hypercube, any difference quotient of the form :

I [ f (PA)--~(P!~)]/(P,',-pja) I 5 2Mld ,PA < P!: where d is the smaller of b! -p,iQ and p i

- ad.

problems in the cnlc~ilusetc. Next, Jhh is still weakly quasi-convex and of class C" so that (3.2) llolds. Then, frolo the co~ivexityin 5 for each I, for instance, (3.8) holds \vit,h A& = fhhp"p). Si~lcef satisfies a ul~iform Lipschitz condition near p, we see that the Aihj are u~lifor~nlybounded as h 0 so a seqile~iceof 8 - 0 ~2111 be cl~osellso that a11 the A;thj tend to limits. Olearly (3.8) holds in the limit. Since tile unit vector in the pi direction is of form 1, t j , we s e e that A; =f p j if j' is of class C'. The last statement follows from ,theorems 011 convex functions. We now define a sufficient condition for f to be (strongly) quasi-convex. THEOREM3.6 : A stcflcieiit cosditiou l o r f to be qucrsi-convez.is tlrut for eitch p tibere exist alternating forms

-

Q

(in which the coefficients ere 0 unless all the a , ...a, are distinct and all the j , ...jP are distinct and an intercllauge of two a's or two j's changes the siga) szccl~ that Jbv ull n we lurve

Proof: For snppose p is any constarlt tensor, G is any bon~~ded domain, a ~ i d[ is ally Lipschitz vector whicll vanishes on dG. By extending 5 = 0 outside G a ~ l dapproxi~ni~ting to it on a larger domain D wit11 srnootll bouiidiwy with fui~ctiousof class C" which vanis11 on a ~ ~near d dD and using Stokes7 theorem we see that the integral of the sum on the right in (3.9) is zero. We now exhibit two iuteresti~~g cases where the weak quasiconvexity o f f ilnplies its quasi-convexity. THEOREM3.7 : Ij' f ( p ) is weakly quasi-oonvex and

tlten f is quasi-convex (1791, [45]). Proof: For, if 5 is Lipschitz and vanishes be assunled smootlr), then

011

dG (which mzly as well

CFIARI.ES B. MORREY Ju. : Multiple intsgvai If me introdace Fourier trltnsforlus (see [79])

we see that

since the i~ltegrandis 2 0 for each y. THEOREM3.8 : If N = v 1 and

+

where P i s continuous and

...,

Then f i s quasi-couvex ht p if and only if P i s convex iw (4, Xy+l). We omit the proof which is fouud in [44]; P is there ~eqhiiedto be lio~nogeneousof the tirst degree in X but this is not necessary in the proof.

problems

(11

the enleulzts ete.

The differentiability of the solutions of certain variational problems with v = 2. 111 this chapter we disciiss the differentiability of the solutions of certain proble~nswhose existence wits proved in 5 2. To save time, me shall not discuss the coutinuity an the boundary bnt shall consider only the differentiability on the interior. This work ww,s first presented in [42], chapters 4,6, and 7 and was the culmination of s series of papers on this subject by Licl~tensteill[34], [35], Hopf (271, imd the writer [39]. Some of these results have recently been generalized by De Giorgi [lo] and Nash [49]. Sigalov [Gl] t~nnouncedresults siinilwr to those presented here. We begin with the following lemma which has a proper'generalization for all values of Y (see [42] and [47]): LENMA4.1 : S'uppose a vector z (x) E 9, O N n domain O and suppose that

for O < r < n ,

(4.2)

1 s (2,)- s (xi) I 5 Ci ( I ) . L . ([xi- ~ ~ l / afor) ~ 0 1 xi - xg 1 5 a ,

where Ql(A) = %1-"--1/2

1-1

,

for every pail. of points (3, xz) in O such that every point on the segmeirt joining them is at a distalbee > a from d B . I'roof: We note first that if E is on the segment and s 5 a ,

j

1 V z (y) 1 dy 5 dl2La-bl+',

B(P,8)

CHARLNSB. MORREY J R . : Multiple ilztegrat

128

using the Schrarz inequality. Next we write

+

I 2 ($2) - 2 (xi)I 5 I (4- 2 (x,)I I ($1

- 2 ($2) I

:

6,

,

+

and then average with respect to x over B r/2) = (xi x2)/2,. I f given t 0 < t < 1, we set y = xk t (x xk), then y ranges over for a B [(I- t)xk t i , rt/2]. Then

+

,

+

from which the result followa. NOTATION: If z€CM2 on G , we define D ( z , G ) =

called the Divichlet integral. LEMMA4.2 : #uppose n E q2on B (so, a) and suppose

where

.

onnuerges, for every .function 2, = z on dB (xo,r ) Thehell

and the right side tends to zero with r.

IVz12dx; this is

problems in the calculus etc.

.

,

Proof: Let p ( r )= D [ Z ,B ( x , r)] Then p is absolutely continuous. For almost all r z (r , 0 ) is AC in 8 with I z, ( r ,'0) 1 in For such r , define

,

4.

Using Fourier aeries, one easily sees that

/" 1

2 (r 9

0 ) - ;(I.)

l2

5

/" 1

2,

,

( r 0) 18 d0 5 r pf ( r )

By computing D2 [Z,, B (m,, r)J and using (4.5) we see that

from which (4.4) followa easily. In order to see that the right side of (4.4) tends to zero with r , we note that

,

THEOREN4.1 : Suppose f ( 8 ,z ,p) is continuozls for all (3: z ,p) alta is convex in p for each ( 8 ,z) , asd suppose there are oonsta?zts 118, M and 1c such that

.

,

,

for all p Suppose I ( # , U) is finite, Q is a bounded domain, and z, miaimizes I ( z , U) among all z in 93%coinciding with x, on 86. Then z, satisjies (4.1) and (4.2) on Q with

I s s / 2 M and L 2 = D [ z 0 B ( r x : , , a ) ] + 2 k n d / M . Tlbzcs 2, satisfies a unifornt Holder co~rditionon each colitpact szcbset of 6 . Proof: Suppose B (a, r) c Q and let Z, be ally function in 33, on B (so,7') and coiuciding with ' z , on d B (x, v) Then, from (4.7)

,

,

, .

,

,

I,ID[z, B,.] - kn r2 5 I (zo Bv) 5 I ( z r B,) 5 M D (2, Br)

The result follows from Lemma 4.2.

+

v2

For the remainder of this section, we shill1 asstune that f ( x , z , p ) sa. tisfies the following condition in addition to (4.7): GENERALASSUMPTIONS: W e assu?n,e that B i s a bounded do$)iain,f satisjies the conditions of Theoreni 4.1, and (i) f i s of class 0" for all ( x , z , p ) (ii) there are functions m, (R) M, (R), and n12 (R) with 0 < vr, (R) < M,( R )for all R 0 such that -

,

at, (R) I n l2


for all ( z , z , p ) such that I X ~ ~ + I Z ~ ~ < R ~ . THEOREM 4.2 : Suppose f and B satisfy the gene~alsssu~ttptions, zo satis$es the contisuity co~~clusions of Theore?,& 4.1, und 5 i s any Lipschitz function on B which vanishes on and near d B , and g, (A) = I (zo At) Then ~ ' ( 0 ) ecists and

+ .

(4.1 1)

P' (0) =

I G

Ifej

[x ,Z" (4,Po (415j (4+.fp;[x , z o (2),Pa ($11 nil ax

Proof: Let I? be the compact support of 5 . Since 8, is continl~ouson I?, 1 x 12 I zo (x)l 2 R2, for some R, for all x on P. Then, for almost all x on F,

+

where, for instance,

Clearly all the Aqg, B;,, and C, me measurable and we conclude also from the general assumptions and the Lipschitz character of 5 that 9 (A) - g, (0) - A

I

(f,j

5j

+fpi

71):

dm = A2 K (1)

a

1

.

where K(A) is ilniforlnly bounded for 11 5 1 The result follows.

problems in the caloulus etc. DEPIN~TIoN: If Q, (0) = 0 for every 5' as in Theorem 4.2, we say that z, fi~rnishesa statiosary value to the integral I ( # ,G). COROLLARY: I f f , 0 , and a, satisfy tke conditions of Theorem 4.2 and if a, mit6imizes I ( z 0 ) atnosg all rufjciently tear z (& se138e) having the same boundary valuea, then zo furnishes a stationary value to I ( a , 0 ) . In order to obtain further differeutiability properties of the solutions z,, we must consider the solutious u of equations

,

where all the coefficients are measurable and satisfy

We begin by considering the case where byk = cjk = 0 and set

Prom our gei~eralassumptions, we see that

Prom thie result and the Poincare inequality (Theorem l.ll),.we see that the space %, is a Hilbert space if we take I, (u , v ; G) as an inuer product sud that the resultiug uorlri is topologically equivalent to the original q2 norm on qz0. LEMMA 4.3: If 8 is any set of j d t e measure, then

Proof : Obviously

I

1 x - X , P-2 dx

9

B!J~P?~)

CHARLESB. M O I ~ I ~3n. E Y: Multiple iwtegrnls

.

.

for every circle B (x, , r ) l'l&en u f E k?, on G and ~ a t i s j i e s

f w a y be tensors. P r o o i : The proof for the general vector u in %*, will follow from the result for class C' whicll vanishes near d G Let z, E G sud sqpose G C R (2, R) and txtend zc to be zero outside 6 . Then if we set v j ( r , e ) = u j ( x ; + r c o s e , z ; + r s i n 0 ) , we see that zc and

.

,

Hence


J

lr-.l-l.

/ I j - ~ x ~ l .I,..(olaraz.

B(m0,r)n G G

Applying the Schwarz inequality judiciously to (4.18), v e obbiu

problea,~ i n the ouloulzis etc.

Using Leltllna 4.3 we see that

Next, define,

From our ass~~lr~ption on f , we see that

Accordingly

The result follows from (4.20)and (4.21). LEMMA4.5: Suppose u and f satisfy the lbypotheses of Le~iuita 4.4. Then fir2 E 2,on G crlzd

Proof: This follows from two applications of Lemma 4.4. THEOREM 4.3 : Thew is an no > 0 a.nd depei~ding o d y on P ~ I , N, M 2 alrd 1 suoh that if 0 < a 4 a,,and H (zo a ) c Cf t h s ~

,

,

11)

,

, , ,

I [u u ; H (x,, a]2 2 D [u , B (xo, a)] for all u E %20 on B (xo, a ) 2

.

CHARLESB . MORBEY Jn. : Mutliple integral

Proof: For

I [u t~ ; H (so, a)]= ro11z ,n ;B (a,, a)]

,

+ J (2b;k

uk

+ cj, ua uk)ax

B(XO@)

,

> U2[U B (so;a)] -

. [1)1, - 2 c:'

&/2

g'"/2aA-p'2

-c

jl 2yfi(121-p],

o<~
using Lemma 4.5 and the Schwarz inequality. THEOREM4.4 : I f 0 < a 5 a,, B (m, a) c G ,byb, cik, a#ad f saiisfy (4.15) and e E 4 on B (x,, a), theye exists a u~tipueu i n CM,, ow B (x, a) such that (4.13) holds for all v E g2,0% B (8, ,a). Xoreover

,

(4.22)

,

B [u B (x, , a)] 21'11'

[I1 e llL

,

+ 0, (1,y) .M2. a"' , 2, < ,u <

(1.

Proof: Froin Theorein 4.3 and the Poincare inequality (Theorem l.li), we see that the space CiO,, is a Hilbert space if we introduce I ( %v), as inner product. Since the equation (4.13) (C3 = B (x, a)) cau be written

,

and since L (v) is a linear fiinctional, we see from Hilbert space theory that there is a unique u in 33,, which satisfies the equation. If, now, we revert to (D[ u ,B (x,, ~ ) ] j las / ~ norm, we see from (4.23) and Letrima 4.4 that the norln of L (v) is given by the bracket on the right in (4.22). The inequality (4.22) follows by comparing the I and D norm. We can now prove the iuterior boundedness theoreln : THEOREN4,5 : Suppose u. E L?,on B (x, a ) c G where 0 < a 5 a, u E q2 on B ( x o ,r ) and (4.13) holds for each v E q2,on B (x, r ) l o r each r with O < r s a . Then

,

,

,

(0,= miu 0,(1,p)) O
5' (x)= h [(Iz - xol - r ) / ( R - r ) ] ,vi = t 2 u j , U j = 5uj.

probleins Then v and UE %,

in

the calculus etc.

on B ( x o ,R). Snbstituting i n (4.13), we obtain

where 111 77 111 is the qo - D-norm % I I ~ (1 u 1 ) is the nonn. Since (4.24) holds for all R < a , the result follows. LEMMA4.6 : If u E bz 010 R (x, R), there is a 21, E %%,otc B ( s o 2R) , such that u, ( x )= 21 ( x ) 018 B (x,, ,R ) attd

,

zohere C4 is atc absolute constant. Proof: Define th2 (8)= 26 (0) on B (3, R ) and extellrl it b y reflection i n the circle B (x, R). Then 11 E 9, on B ( s o ,2 R ) and

,

,

Then, define 21,

(x)= 1~ [( I x - $0 1 - R)IR]*

( x ),

where h is filnction i~~troducedi n the proof o f Theorem 4.5. Then u , is easily seen t o have the desired properties. T H E O R E M4.6 (Dirichlet growth theorem): Suppose 0 < @
,

I

,

I e ( 2 d x < L z ( r / 8 ) 2 ~ ,O ~ r < d = a - 1 x , - s o l ,

B(Xl,).)

.for some ,u with 0 <,u < R/2 attd m,/2.!1, and every circle B (xi ,r ) c B (a,, a). l'heti 21 satisfies the cottditiou (4.1) attd (4.2) with G replaced by B (so, a ) , xo

CHARLESB. MORRIGYJR. : Multiple integral

'

,

replaced by xi, a by 8 = a - I x , - xo 1, R replaced by ,u and L replaced by Cg, where C5 depellds ouly on mi M i,M2 L 1,p ,a , asid Ill u 111 where

,

, ,

J B(xo,~)

,

Thus u satis$es a uttifornorw Holder condition on n7q B (x, R) with R < a which depends only 0% the qtbantities above and a - R. Proof: Let ,lJ?-- bb" ,X uk 6;, pj = b;j 2&+, cjr ZC' f,

,

+

+

, , , and f's

From our hypotheses on the b's c's e's 4.6, nnd 4.6, we see that

and from Lemmas 4.4,

Moreover u satisfies the equation

,

,

(4.26) I, [U v ;B (xi r)]= -

I

( B y v?

+ Fj vi) d x j

v E q2,ou B (gi ,v)

B(xI,~)

,

on any B ( x , r ) C B (x, ,a). As in the proof of Theorem 4.4, there is a unique solution U, of (4.25) which is in CM,, on B (xi r ) and

,

where 2, depeuds only on the quantities meutioned. Now V , = u - Ur satisfies the homoge~ieous equation (4.25) and so clearly lninimizes I, [V , V ; B (mi r ) ] among all V = V,.(= u ) ou d B (xi r ) . Since Ur E %20 on B (xl r ) me see that

,

, ,

, .

where B , = B (xi r) Using the fact that I,,( V, any u, = 2~

, V, ;B,.) 5 IQ(u, ,u, ;B,)

on d B (2,,r ) and using (4.16), we see that

for

problems in the calculus etc. where Z2 depends only on the quantities indicated. The results follow from Lemmas 4.2 and 4.1. We call now resume our discussion of a solution x, of a variational problem of the type beiug discussed here. THEOREM4.7: Suppose a, gives a stationary value to I ( a , G) and satisfies the continuity conclusions of Theorem 4.1. Then a, E C l + p on ecr.ch domain with F C 6 where 0 < lu < 1 and the derivatives E %if on domains inte. lior to 6 . Proof: Since y (0) = 0 we see that the right side of (4.11) holds for each Lipschitz 1' with compact support in G So, suppose B (so,a) c O Choose d > a so that B (8, A ) C (S Then, from Theorem 4.1, we have x la 2, (x) < Re, for some R ou 8 (x, A). Let b = (2a $ A)/3 c = (a 2A/3), h, = ( A - 413, let e, be the unit vector in the $7 direction for y = 1,2, let v be ail arbitrary Lipscl~itz function having support in B (GC) and define

r

,

,

,

I

,

+I +

[i,(x) = h-1

,

.

.

.

,

,

- d (x)],uj, (x) = h-I [z{ (x + he,) - z j (x)] ch has support in B (x, , A ) . Substituting Ch into

[vi ( x - We,)

for 0 < 18 I < h, Then equation ~ ' ( 0= ) 0 and usiug (4.11), we see that on R (x, ,c) with coefficients a;$, etc., where

.

zch

the satisfies equation (4.13)

for almost all .x. From the general assumptions or1 f and from the for~nnlaa (4.26) for the coefficients, we see that the bounds (4.14) arid (4.15) hold uniformly for 0 < I h < h, with

1

m,=m,(R),Y,=M,(R),M,=KMz(R),21=m/M, G = B(x,,c), where K is a constant depending on 1 and the distance of B (x,, A) from

,

8 0 . Clearly each uh E on B (xO c) and its L, norm is uniformly bounded there, and we also have

1

I e h l ~ ~ ~ o~ <; I P I <, ~ , .

B(ajr)

Accordingly, we see first from Theorem 4.5 that the 9, norms of the uh are uniformly bounded on B (x, b) and then from Theorem 4.6 that the uh satisfy e uniform Holder condition on B (ao a) independently of h Thus we ma.y let h 0 and we see that the derivatives ~;i,E %; and satisfy this

,

-

, .

Hiilder condition on B (8, a)

,

.

CHARI,ICS B. M O ~ ~ R ~JR. C Y: Multiple itltegral

CHAPTER V

A variational method

ill the theory of hrrmo~ticirrtegrals.

In this section, we apply our variatio~lalmethod to the study of armonic integrals aud, more generally, use it to obtain tlle Kodaira decolupositiou theorem [29] (see Theorem 5.10 below). This approach was origit~ally suggested by Eodge in his first paper on the subject 1251. The generality of the manifolds allowed and the methods and results obtaiued are closely related to those obtained by Friedrichs 1201 working intlependently. Of course correspot~dingresults have been obtained on smoother ma~lifoldsby a number of other authors using otlrer methods ([12], [23], [26], [29], [38]). I n this section, we sllall coufilte ourselves to compact mauifolds without bonndary. The variatio~talmethods a.re applied to compact manifolds with boundary in [SO] nnd [46]; boundary value problems for forms have been coltsidered by other writers using bther methods in [13], [66]. . We adopt the usual definition of a compact Riemannian manifold of dimension n (instead of v ) and of class C b r . Cf: (0 < ,u 5 1) ally two admissible coordinate systems are related by a transformatiou of class Ck or CF, respectively. If 0 <,u< 1 , the class C : is the same as what we have called C k f r ; If ,u = 1 , a function is of class C: if aaltd only if its derivatives of order sk satisfy Lipscllitz conditions; transformations of class C: are defined similarly. If a coordinate system is of class C i , the illduced gijare of class Ck-1. W e shall assume that our ~t~anijololdis of class at least C:. We shall be concerned with exterior differential forms of degree r on a mttiifold M ; we call these simply r-forms. I n the domaiu of a given coordinate system such a form o may be represented by

where mioi,...ir are the components of cv in that coordinate system and A denotes the exterior product. I n order to take care of the case of non-orientable manifolds, me allow both eoen and odd forms. If two coordinate systems (8) and (0') overlap, the components transform according to the law

+ 1 for even forms,

J/I J I form odd forms,

J=

a (g6,..gn)

8 ('xi, ,.. ,I$*) '

problents in the calculus stc.

Since the Jacobians involved in (5.2) are at least of class C; (Lipschite), we may say that a form w is of class 4 or q-its components in each coordinate system are. Given'an r-form w , we define its dual +w by

+

.where epl...,n is 0 if two indices pi are the same or otherwise is 1 according as pl ...p,, is an even or odd permutatio~~, ki < < kr .are chose11 so that k, k, jl j,-,. is a permutation, I'(k)(l) is the determinant of the

...

...

Ic 1.

,

6

..,

r

g i j aud I' = _+ chosen so that hiA ... A hlI = d B , tile positive volume element. I f two' fomls o and q of the #(,me lci?cd (both even or both odd) of the same degree are Q J2on M , we defiue their inner pvodtcct

we form inner products only under these conditions. If o is an v-form given in the x-system by (5.1) aud if g is an s-form of the same kind with a corresponding representa,tion, we define

Accordingly the inner product (a,, q) is also given by

...

where (i)= i1 i,', where il <

... < i, , etc. In case P corresponds to x,

a: system and gv(x,) = 6 i j , we see that

in the

CHARLESB. MORRICYJR.: ikultiple integral

The following theorem is well hoown and is evident. THEOREM5.1. For each r = 0,1, ... n the totality of r-fornzs of a $xed kind J2 on M (with epuivrrle~itforgns idesti$ed)ji)~aisa retrl Hi1bei.t space 2; with i n ~ t e rp~.oduct yioerl by (5.4) Iu order to introduce ~ L I Iinner yrotluct ill 3;j, on M me proceed as follows : DEFINITION:Let % = (Ui U Q ) be a finite ope11 covering of M by coordillate patches Uq = Qq (B,),where each Q, is a Lipscl~itzdomain in P . If w and 17 are in %, on M we defir~c

,

,

,...,

where w(q) and (6)

178; s.re the

compollents of

OJ

and 7 iu Q , . Then

is the expression for the norm in q2on M corresponding to the inner prodtlct (5.8). I t is char that convergence of wl, to w according to one of the norms (5.9) is equivalent to the strong convergence in %, of the cornpenelits wk in any coordirlrbte system to those of w . Thus we obtain the theorem : THEOREM5.2: For ench coordintste covev %! and each r = 0 , la the space of r-forms in 9, of a given kind on M forms a real Hilbert space 9;with i n ~ t e rproduct given by (5.8). A n y two sz~chinner prodttct sare topologically equivalent. Now, if w is an r form E q2, we define dw and Bw by

...,

6w = (-

l)l+M'-l)

+ d,r; w, and

+

We note that d o is an (r 1)-form (if r 5 la - 1) and 8w is an ( r - 1)form (if r 2 1 ) . Finally, me define the Dirichlet integral by

THEOREM5.3 : d i s a bounded operator frolit the whole of q,' into &;+I, and B is a boz'nded operalor jrom the while of %: into 2;-' j each of these operatola preserves suetmess or oddsess. .D (w) is a lower senti-continuous functios'with resped to weak convergence i j r %;. If ol, tends weakly to wo i n

9; on M

, then wk tends strougly

to w , i n

2; oa M .

problem in the calculus etc.

Proof. Tlle first statement in clear form (5.8) since the gii are at least Liyschitz and have bounded first derivatives. Ncw if wk tends weakly in 33;3 to w , dok and 6wk teud weakly in to dw and 6w, whence the last statemellt about D (w) follws from the lower~semicontinuity of the ~ i o r ~inn J8 with respect to weak couvergence. The last statement is an application of Theorem 1.13. From (5.6) and (5.7), we see that

I n the coordinate system of (5.57, we see that

where i, < ... < i,. and i, we see that (5.14)

... i,j, ...j,-, is a permutation. Trom the form (5.12), xx

w = (- l)W--r) (,j

.

From (5.5) and (5.10) it is easy to see that

where q is any s-form (and w is an r-form) in Prom the rules of exterior multiplication and (5.5), it is easy to see that (5.16)

q ~ w = ( - l ) ~ ~ w ~ q .

Prom (5.4), (5.12), (5.14), and (5.16), one derives

,

If M , w and 5 are a11 smooth and w and and degrees r and 1. - 1, respectively, we obtain

5 are

of the same kind

CHARLES B. MORREY JR. : Mtdtiple integmt

-

since the Brst integral vanishes by Stoke's theorem for (n 1)-forms, the bracket being just d [x w v 51 (see (5.15)). We emphasise the result: ( d o , 5) = (6, d5)

.

In the case of smooth manifolds and forms, we see from (5.10) and (5.14) that (5.19) Combining this with (5.18), we see that

The formulas (6.18) and (5.20) can be extended to 332 forms on manifolds only of class C : by using a proper partition of unity (recall Lemma 1.3), such that' if the supports of two of the hi intersect then their union lies in one coordinate pat&, to represent each form as a sum of forms whose supports have the same property. Then, for instance

,

(do 5) = 8 w

58)

and each term may be evaluated using one coordinate patch ; in that patch, the gij and the forms may be approximated by smooth forms. In the case of a coordinate system of the type in (5.7) where we also assume that all the dgij/dxk = 0 at q,, we see from (5.10) and (5.13) that the components of d o at x6 are

...

...

where il i,-l 11 l,-r+l is a permutation. From (.5.21), we see that Diri. chlet integral D (w) in (5.11) reduces to

for the case that o has support in a coordinate patch having domain (7 and the g,, = Bij throughout G ; the last integrals all vanish in this case.

problem in the calculus

stc.

We now prove the followiug important lemma, first proved for forms by Gaft'ney LEMMA 5.1 : Given E > 0 0 r 5 n and Po on M, there is an adwzissibile coordinate system mapping B ( 0 , e), for some e > 0 , onto a ~zeiglbborhood U of P o , and a constant 1 such that

,

,

J

2 D ( w ) ( 1 - E) (i)a B(o,e)

CO?~),,

,

dx - 1 (w w )

332

for any r-form E whose support is in U. Proof: We begin by choosing a fixed coordinate system mapping some BR = B (0, R) onto a neighborhood UR of Po,carryng the origin iuto Po, and satisfyng go ( 0 ) = d i j . From our formulas for dw and 6 0 , we see that (5.24)

+

J

D (o)= [a(W)aBoci),tz o(j , , ~ 2b@)( j l a o ( i ) , aO(j)

+ ~ (( f0l wci,w(

j)]

dx

Be

where the a's are combination of the gu only and so are Lipschitz and the b's and c's are combinations of the gij and their first derivatives and so are bounded aud measurable at least. Since the a's are Lipschitz and since

me see that we may choose

Q

so small that

The result follows from (5.22). The following important theorem corresponds to Garding's Inequality for differential equations : THEOREN5.4 : For each r = 0 , n and coordinate covering C2e of M , there exist constants K y > 0 and Lg such that 0

,...

for ever w E %. Proof: Prom Theorem 5.2 it is sufficient to prove this for some particular q.Let ?e = ( U i UQ)be an opeq covering of M by coordinate 1 patches such that ench ,c€ M is in some Uk satisfying (5.23) with 8 = 2

, ... ,

'

CHAR^.^^ B. MORREY

, ... ,

JR. : Multiple integral

say. Let Cf, U Q be the domi~in in E" such that Uk= Qk (Gk) for a11 k . There exists a finite sequence @, ,a,of Lipscbitz functions on ]I[, each of whioh has support interior to some Uq, nnd such that

, ... ,

for all x E M. Now if (5.25) aere false for the Q just described, there would exist a sequence (up)of $'-for~usill %; 811ch that D(up)aud (up, cop)mere uniforlnly bounded but 11 0.1, l r r - w Then, for some s p and some subsequence, still called u p ,me mould have

, ,

.

mhere @, has support in Uqt since

and

But it is easy to see that D (@, w,) and (as u p ,QS op)are uniformly bounded. From our choice of neighborhoods we have reached a contradiction with the fact that

We can now present the variational method. We begin with the following lemma : LEMMA6.2 : Let % be any closed linear manifold it, the space 2;of y;fovnts on N (of some one kind). Then eithev there is no f o m o of C)j7- which is in % or there is a form o, in %%; with ( m , , o,)= 1 which minimines D (o)among all such fornts. A ~ o f If : 972 contains no form in %;, there is nothing to prove. 0. thenvise let be a minimizing sequence, i. e., one such that ( a kmk) , =1 slid okE 9 ' 2Il9; for each ?c = 1, 2 and such that .D(ak)approaches its iufimulu for all u € % n %:. Prom Theore111 5.4 it folloms that the

,... ,

problenb in the calculus etc. ((wk,wk))ze are uniformly bounded. Accordingly, a snbseqnonce, still called (wk), exists which converges weakly in %' to some form w,. But from Theorem 5.3 wk tellds strongly in Pi to w, and 7) ( w ) is loaer-semicontin u o ~ ~with s respect to weak convergence in 93;. The proof of the lemma is now complete. DEFINITION:A harntonic yield w on d l is a form ill CM, on J4 for whicl~ dw = 6 0 = 0 almost everywl~ere..We mill let W denote the linear manifold of harmonic flelds on !I. of degree r . (Strinctly speaking we have %[ and for even and odd fonns, respectively). THEOREN5.5 : For each 1. = 0 n ( = din M) the littear lnaizifold W i s finite dimensional. Prooj. The 932 forms are dense ili E l , since the Lipschitz forms are. Let MI = 2;.There is a form wl in MI ll 93; which minimizes D (w) among all such forms with (ci, 6 ) = 1 Let $1, be the closed linear manifold in 2; orthogonal to w l , a l ~ dlet wz be the correspo~tdillg miliimizing form in M , . By continning this process, we may determine successive minimizing forms o, w,, w, , each satisfying (wk wk) = 1 and beillg orthogonal to $111the preceding ones. Now if ll (mi) > 0 , there are no harmonic fieltls 0 since D (w,) 5 ID (w,) 5 On tue other hand, suppose D (wk)= 0 for all values of K . Then by Theorem 5.4, ((wk wk))?( is uniformly boullded in k whence a subsequence [ a p )converges weakly ill 93; and hence strongly in 2; to some form wo in %;. This is impossible since the wk forrn an o~thonormal system in 2;. THEOREX 5.6: For each coordinate coveiing % of M there i s a oo?istntzt A, stwlb that

, ... ,

.

,

,

,

... ,

....

+

,

,

for any w in %; wich i s orthogolaal to %' . ProQf. For, let wo be that form in %; (there is one si~lceeach harmonic field is in q2) which minimizes D ( o ) among all o i l l 9; wit11 (w ,w)= 1 and o orthogolial to W . Then clearly D(w,) > 0 and by hol~iogeneity

for all o in %; and orthogonal to

from which (5.27) follows.

qr.By Theorem 5.4 we see that

CHARLESB. Mottss~Jlt.

:

Multiple kbteyral

TnEoREM 5.7: Suppose coo is any forwt in L?; crfrd ortlrogottal to 9[r Therb there is a uttigue form Qo in 9: and ort1bogollal to qr such thut

.

(a Qo, a 0 -t @Q0,dr) = ( 0 0 , 5) for every C in 93;. Moreover, the tratt.pformation front wo to Qo is a bou~tded linear tmnsformation from &; into Proof: From Tt~eorem5.5, we see that

.

,

since (w wo) is a bouilded linear fuactional on %; here 11 w =((w,s))qE. Hence I ( w ) is bounded below and is lower~semicontii~uo~~s with respect to weak convergence in %; if w is orthogonal (22-ser~se) to C3e". Accordingly there is id minimizing form Qo. If 5 is any form in orthogoultl to %I' we then see that

,

which shows that (5.28) hold's for all such 5 and Qo is nnique. But then (5.28) holils all 5 in since any such 5 is uniquely representttble in the form 9 = Il+ 5, where dH = 8H = 0 and to is in 9; a11d orthogorial to q v . Finally, if we set [=no in (5.28) aud use Theorem 5.7, we see that

from which the last statement follows. DEFINITION : The form Qo of Theorem 5.7 is called the potential of coo . We observe that if all forms in (5.28) and the mallifold M were sufficiently smooth, the equation (5.28), together with equation (5.18) would i ~ a ply that

I n any coordinate system, (5.30) reduces to a system of second order eqoations iu the components of the forlus; if r 2 1 , these equations involve the second derivatives of the gij as well as those of the components of Do. However, all the results stated so far hold for mauifolds of class 0;in which case the requisite second derivr~tivesof the gq certainly do not exist.

problem in

tH8

caleultrs etc.

,

DEFINITION: We say that o i s of olass 0 5 1< 1112 if for each coordinate system 0 with domain B R , there is a constant 1, = L (8,w ) such that

The olass 3321is defined si~nilarly. The importaucd of the spaces %Izn ariees from the fact that if w E ()32n with I = p - 1 n / 2 , O < p < 1 , then w E C; ; this follows from the stminghtforwand extension of Lemma 4.1, to n dimensions. We can now state the following results concerni~~g differentiability. THEOREM5.8 : 8uppose that w E &; @ %' and 9 i s its potential. (i) If M i s of class G: , the Q ,dQ , and 8Q ;2 E2 (ii)I f M ie of cla,ss o:, and m E g2,,then Q , dQ, and 8Q E q 2 n and hence in 0: i f 1 = 1112 - 1 p 0 < p < 1. (iii) I f 42 ie of class 0: and o E $32, then dQ a i ~ d8Q are the potentials of d o and 8 0 , respectively. (iv) I f 44 i s of class C; and uj E c:-' k 2,O < p < 1 then 9 ,dQ and 8 8 E I f 16 2 3 and w E O;-\ thetc Q E CE-l. (v) If M and m a l e of olass Cm OY analytic, then so i s P. I n all case, if we set a = d 9 and /3 = 8Q we have

+

.

+ ,

O F.

,

,

T~EoREM5.9: 8uppose that H i s a havi)&onicfield. (i) I f dZ ;2 C: , them HE CM2a with 1= n/2 - 1 p for any p 0 0) I'roof: Obvio~islyH satisfies (5.28) with w,, = 0 . Then equations (5.28) are a special case of the more general equations

+

,

.

Using (5.24) and (5.22) we see that equations (5.32) are equivalent to equations of the form (4.13), if l has snpport on some one coordinate patch, where the a's are Liptschitz, the b's and 0's are bounded and measurable,

aud the e ' s and f ' s EEz. Such systems have bee11 studied extensively by the writer in [751 and [47]. Since Professor Nirenberg7s lectures are concerned with differentiability problems, tile resl~ltsand their proofs are omitted. The results c o n c e r ~ ~ i8~ ~and g H follow directly from the result just mentioned. To prove the differe~~titibilityof d 8 and 8 8 , we select a coordinate patch and find t l ~ a twe can approxitnate to 8 , 0 , aud the gij by smoot functions so That JZ is a potential of w with respect to the altered gij t ~ teach stage. Then, if 5 has support interior to this patch, we see thnt (5.31), (5.18), and (5.20) imply that a and 1 satisfy (da

,d l ) + (8%- o ,dt) = 0

The iuterior boundedness theorem (like Theorem 4.5) ttud an approximatioli theorem for such systems allow us to pass to the li~nitill (5.33). If we use (5.33) and (5.18) to see that u and g are the potentials of dw and 40, respectively. The following theorem complements the well-known orthogonal decomposition of Kodaira [29]. THEOREM 5.10 : I f w is ally for*), i ~ &, t the,, there exists a 8armolzic field H a d forms a f l , atid 8 i n q2such that

,

where 8 is the potential of w - H. I f the .first equation of (5.34) holds for a harvtonic field H, and forws a, a ~ db, i n 33,, thew Hi = H , da, = 8a at&d ag, = ag and (Lf of a11 for*)ts dB for The sets or all forms 6a for a i n B i n 9;-I are closed linear manvolds i n 2; and

.

,

e'

,

I f M E 0: altd 0 E J2,4or %2A , 0 I IZ < n/2 the,&6a atcd dB have the same properties. ~f d . l € $ and w ~ ~ with f , 1 ~ 2 2O, < p < l , O < o < l , a t ~ d e i t h e r l < k - 1 or 1 = k - 1 and asp, then 6~ and dB have the same d(Teretttiabi1ity properties as w . I f Y and 0 E 6'" or are altalitic, so are 8a and dB.

problems in the ealculua etc.

Proof: The first statement and the. differentiability res~ilts follow immediately from Tl~eorems 5.8 and 5.9 If H a , and fi all E % (ai~dhave properly related degrees), formulils (5.18) m d (6.20) and the defiuitioi~of harmonic field imply t11at H , da and d l are orthogo~~al in J2. To see that the sets e r and W are closed we see, by following the constructioil in the first paragraph of the theorem with o = da ;tud dB in turn, thirt if a and B E q2 there are forms a, and 1, ill q2~11dorthogonal to % such that

,

,

,

-

Then if Ba, o in J2,we see that the al,,- some a, in.W2 by Theore111 5.6. A corresponding result holds if dln -z in J2

.

BIBLIOGRAPHY

[ I ] N. A R O N ~ Z A Jand N K. T. S M I T H2runotional , spaces and functional completion, Ann. Inst. Fourier Grenuble 6 (1956), 125-185. [a] F. E. B~towulen,Stro1,gly elliptic systems of diffcrestial equatiore, Contribntions t o t h e theory o f partial differential equatio~ls,15-51. Ann. o f Math. Studies, No 33. Prinoeton University Press (1954). Numerous notes in Proc. Nat. Acad. Sci. U.S.A. 38 (1952), 230-235 131 and 741-747 ; 39 (1953), 179-184 and 185-190 ; aud Inany others. [4] J . W . C A L K I NPu1106i0128 , of eeveral variables and absolute oontinuity, I, Duke Math. J. 6 (1 940), 170-185. [5] L. C E ~ A I A~ nI ,e i s t m e theorem of the calculee of variations for i~ltegrale on parametric surfaces, Amer. J. Math. 74 (1952); 265-295. [6] S. C I N Q U I N Sopra I, l'eetrtmo aseoluto degli integrali doppi it, forma ordinaria, Ann. Math. Pura Appl, ( 4 ) 30 (1949) 249-260. [ I ] R. C O U ~ ~ A NPlateau's T, problelir and Dirichletle principle, Ann. o f Math. ( 2 ) 38 (1937) 679-724. Dirichlet's principle, co*lformal mappiag, and minimal eurfaoes, 181 Iutersoienoe Press, New York (1950). [9] J. M. DANSKIN,On the existence of minimizing surfaoes in parametric double integral problcms in the calculus of variations, Rev. Math. Univ. Parma 3 (1952) 43-63. [ l o ] E. DE Gro~tor,Sull'aralitidtd delle eztremali degli integrali multipli, Atti Aooad. Nnz. dei Linoei Retld. C1. Soi. Fia. Mat. Nat. ( 8 ) 20 (1956), 438-441. [ l l ] J,. DENY, Lea potentiels d'e'nergie $nil Aota Math. 82 (1950), 107-183. rla] G. DE &HAM and K. KODAIRA, Harmonic Integrals (mimeographed notes), Institute for Advauoed Study, Pdnoetou, N. J. (1950). [I31 G. F. D. DUFF and D. C. SPENCER,Harmonic tensors on lliemasnian manijolde with boundary, Ann. o f Math. 56 (1952), 128-156. [I41 G. C. EVANS, Fundamental points of potential theory, Rioe Inat. Pa~nphlets7 (1920), a52-359. , Note oa a theorem of Bhher, Amer. J. Math. 50 (19%8),123-126. [I51 , Potentials of positive ma88 I. Trans. Amer. Math. Soo. 37 (1935), C161 226-253. [17] G. FICEERA,Esistenza del miaimo i n un ctassico problema en caloul delle variazioni, Atti Aooad. Naz. Linoei Reud. C1. Soi. Fis. Mat. Naz. ( 8 ) 11 (1951), 34-39. 1181 K. 0. FRIEDRICH,On the identity of weak and strong mtsnsione of differential operators, Trana. Amer. Math. Soo. 55 (1944), 132-151. , On the differentiability of the solutions of linear elliptic diferential r191 equations, Comm. Pure Appl. Math. 6 (1963), 299-326. , On differential forms on Biemannian manifolds, Comm. Pure Appl. POI Math. 8 (1955), 551-558. [ a l l G. FUNINI, I1 prinoipio di minimo e i teoremi di esietenza per i problemi di contorno relativi alle equazioni alle derivate parziali di ordine pari, Rend. Ciro. Mat. Palermo 23 (1907)) 58-84.

,

,

problems in the calculus etc.

151

-221 M. P. CAFFNEY,The harmonic operator for ezterior differential forms, Proo. Nat. Aoad. Soi. U.S.A. 37 (1951), 48-50. , The heat equation method of Milgram and Rosenbloom for open 1231 Riemannian& manifolde, Ann. O f Math. 60 (1954)) 458-466. '241 L. OARDING,DiriChletle problem for linear elliptic partial differential quatione, Math. Scad. 1 (1953), 55-72. [25] W. V. D. HODGE,A Dirichlet problem for harmonic functionale with applications to analytic varietiee, Proo. 1,ondon Math. Soo. ( 2 ) 36 (1935), 257-303. The 1 heoly and Applications of Harmonic Integrale, Second Edi1261 tion, Cambridge University Press 1952. [27] E. HOPF, Zum analytischen Charakter der Loeungen regularer oweidiniensionaler Variatiosgrobleme, Math. Zeit. 30 (1929), 404-413. [28].'b J O H N ,Derivatives of continuoue weak eolutioue of linear elliptic equatione, Comm. Pnre Appl. Math. 6 (1953) 317-336. [29] K , ICODAIRA, Harmonic jielde i a . Riemannian maaifolde, A m . o f Math. 50 (1949), 587-665. [ S O ] P. D. I,Ax, On Cauchy'a problem for hyperbolic eqnatione and the differentiabilily of the solutiona o j elliptic equatiot~s,Conim. Pure Appl. Math. 8 (1955), 615-633. [31] H. LEBESGUE,Sur le probldme de Db.iclilet, Rend. Circ. Mat. Palerum 24 (1907), 371-402. [32] B. L R V I , Sul principio di Diriohlet, Rend. Ciro. Mat. Palermo 22 (1906). 293-359. [33] H . LEWY, On minimal snrjacee with partially free boundaty, Cumm. Pnre Appl. Math. 4 (1952), 1-13. [34] L. L I C H T E N S T E I N Uber , den analyticcol~enCharakter dev Liiaungen oweidirereionaler Varicrtioaprobleme, Ball. Acad. Soi. Cracovia, C1. Soi. Mat. Net. (A) (1912), 915-941. , Zur Theorie der Konforme Abbildung. Eot~forme Abbildung nichtana[351 lylioher ehgularitatenfreier Plaoltenetiicke auf eb61ie Oebiete, Bull. Int. Aoad. Soi. Craoovia, Mat. Nat. C1. ( A ) (1916), 192,217. [Y6] E. J. MCSHAXIC, Integral8 over surface8 in paralltetric form, Ann. of Math. 34 (1933), 815-838.

,

- , Parametrizatione of eaddle eur.face8 with application to the problena of Plateau, Trans. Amer. Math. Soc. 35 (1934), 718-733. [38] A. MILGHAMand P. ROSENBLOOM, Harmottic forms a d heat conduction, Proo. Nat. Aoad. Soi. U.S.A. 37 (1951), 180-184 and 435-438. [39] C. B. MOILPEY,Jr. 01~the solution8 of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soo. 43 (1938)' 126-166. , Functions of several variables and abeolute continuity I l l Duke Math. J . 6 1401 (1940), 187-215. , A correotiou to a previous paper, Duke Math. J. 9 (1942), 120-124. 1411 , Multiple integral problems in the oalculue of variations and related topioa, ia21 U u i v . o f Calif. Publications i n Math. new ser. 1 (1943), 1-130. , The problem of Plateau on a Riemannian manifold, Ann. o f Math. 49 [431 (1948), 807-851. , Quaei-convezity and the lower-~emioontinuity of multiple integrale, Pao. J . [44l Math. 2 (1952), 25-53. , Seoond order elliptic system8 of differestial eqaatiora, Ann. o f Math. Studies 1451 No. 33 Priuceton University Preas, 1954, 101-159. A variatio~ial method iu the theory of harmonic iategrals, I1 Amer. J . [461 Math. 78 (1956), 137-170. [371

-

,

[47] C. B. M o n n l c ~and J. Es1.r.8 Jr., A variational method i n the theoq of harmo~lic integrals I , Ann. o f Matli. 63 (19a6), 91-128. [48] M. NAGUMO,U b ~ rdie gleichmiiseige Suntierbarkeit und ihr6 An~uesdung auf eia Varia/ioiigroblem, Ja.1). J . Math. 6 (1929), 173-188. [A?] J. NASI?,Continuity of solr~tions oJ pat*abolic and elliytio equatioae, Amer. J. Math. 80 (1958), 931-954. [80] 0. N I K O D Y MSur , nne clasee de foactions considerden danlrs l'titnde du probldme d s Diricldet, Fund. Math. 21 (1933), 129-150. [ 5 l ] G. N B B E I ~ I N G Uber , die erete Randzoerla~~fgabe bei regffilarm Fariatioitsprobletnen I. Math. Zeit. 51 (1949), 712-751. [52] H. R A D ~ M A C H PUber I ~ , partielle and totale Differeszierbarkeil vo11 Fffinlctionen mehrever Variabeln, nnd uber die l'ra~sformatios der Doppelintegrale, Math. Ann. 79 (1918), 340-359. [53] F. ~ Z E L L I C H , Ein Satz Uber mittlere Konvergence, Qottingen Nach. (Math.-Phys. IC1.) (1930), 40-38. [54] H. SAKS, Theory of the latsgral, Seo. Ed., Warsw-Lwow, 1937; English tr.anelation b y L , c. Yollng. , Oa the surfaces ruitho,ut tangent planes, Ann. o f Math. 34 (1933), [551 11 4-134. [56] L. S C H W A I ~ TTheori, Z, den disti~ibutioaeI st 11, Aotnalites Soi. Ittd. 1091 e t 1132, Pnbl. Inst. Mato. Univ. Strasbonrg 9 e t 10, Paris 1950 ot, 1961. [57] M. S H I F R M A ND~ferertiability , and aealyticity of eolutiolce of double iategral variational probleme, Ann. o f Math. 48 (1947), 274-284. [58] A. G. S ~ G A L O VConditione , for the ezistei~oeof a minimum of double integrals in u r nnbounded 1-egiou, Doklady Akad. Nank SSSR.(N.S.) 8 (1951), 741-744 (Rassian). PI , Regular double integrals of the calcultu of varialione i n non-parametric form, Doklady Akad. Nauk SSSR (W.S.) 7 3 (1950)) 891-894 (Rnaeian). Tiuo dimensional probleme of the cabulus of variations i n floa-parav~ietrio Po] form. transformed into parametric form, Mat. Sbornik NS 94 (76) (1954), 385-406. On conditions 01 differentiability and arialiticity of solutions of two dimenL6l1 sional probleme of variations, Doltlady Akad. Nank SSSR ( N S ) 85 (1952), 273-275. [62] G. I. SILOVA,Existence of an abeolute minimum of multiple integrals i n the calculus of variatioas, Doklady Akad. Nank S S S R (N.S.) 102 (1955), 699-703. ~631 , 2'wo dimessional problems of the oaloulus of variations, Uspehi Matem. Nank (N.S.) 6 (1951)) 16-101 (Russian). 1641 S. SOBOLRV,Sur quelques svi~luationsconoemast les famillee L s /oncfions ayant dee deriveee a aarrd integrable, Comptee Rend. Aod. Sci. SSSR. N.S.I. 1936, 279-282.

,

, On a theorem of functional' analysis, Mat. Sbornik N.S. 4 (1938). 471-497. [66] D. C. S P E N C E IDiriohlet's ~, principle on manifolde. Studies i n Mathematics and Meohanios presented t o Richard v a n Misee, Aoademio Press (1954), 127-134. [67] G. STAMPACCHIA, Sopra una clasee di funzioni i n due variabili. Applioazioni agli integrali doppi del calcolo delle variazioni, Giorn. Mat. Battaglini ( 4 ) 3 (79) (1950), 169-208.. Gli integrali doppi del calcolo delle variazioni iii fornla ordinaria, A t t i PSI Aooad. Naz. Linoei Rend. C1. Soi. Fie. Mat. Net. (8) 8 (1950), 21-16. Siatemi di equazioai di tipo ellitttco a derivate pattziali del primo ordine [69] e proprieth degli eslremi degli integrali mnltipli, Ricerche Mat. 1 (1952)) 200-226. Problemi a1 conlorno per equazioni di tipo ellittioo e derivate parsiali e quePOI stioni di calcolo delle variazioni coanesae) Ann. Mat. Pnrii Appl. ( 4 ) 33 (1953), 211-238. 1651

,

,

-

problems in the calculus etc. [71] L. T O N E L L I Sulla , quadratutr delle aupergcie, Atti Aoond. Naz. Linoei Rend. C1. Soi. Fie. Mat. Nat. (6) 3 (lS26), 633-638. , Sui massimi e miltimi assoluti del calcolo delh variazioae, Rend. Ciro. C721 Mat. Palermo 32 (1911), 297-337. Sul caso regolare nel calcolo dells variazioni, Rend. Ciro. Mat. Palernlo P I 35 (1913), 49-73. Sur une ntdthode direcfe du calcul des variations, Rend. Ciro. Mat, Pa[741 lermo 39 (1915)) 233-264. La semicotttinuit& nu1 calcolo delle variazioni, Rend. Ciro. Mat. Palermo [75] 44 (1920), 167-249. , Fondamenti del calcolo delle variazioni, Bologna, Zaniahelli, 3 vole. [761

-

, ,

,

,

Sur la semi-costiattild den inftfgrales doubles du oalcul den sai.iaUon8, Aota Math. 53 (1929). 325-346. [781 L'eslre~uo aesoluto degli integrali doppi, Ann. So. Norm. l'iea (2) 3 (1933), 89-130. [79] L. VAN HOVE, Sur l'extenaion de la condition de Legetdre du calcul des variations aux integrals multiples a ylusieulr fonctions ittconttues, Nederl. Akad. Wetensoh. 50 (1947), 18-23.

[771

,

CtUBBIO

-

S O Q . !l'Il'O~RAFIClA

~ O D E R I S L1~9 6 0

k~tl.tbi1~0 diagli dtnmii dslln 8o~ol~olnNornbnis &?peviove di Pisa Revie TIT. Vnl. XIV. Fmc. IV (1960)

UNIPORMIZZAZIONE E MODULI (3 di L. BERS (New Pork)

Le equazioni di Beltrami a coefficienti discontinui furono considerate per la prima volta da Morrey (cfr. [3] anche per i riferimenti bibliografici); ease si sono dimostrate utili nello studio dell~uniformizzazionee dei moduli delle superficie di Riemann. Nella teoria dei moduli che se ne deduce, ci si B limitati naturalmente a1 B caso classico B ; tuttavia alcuni teoremi posso110 essere dimostl.ati a~ichepel caso di superficie di Riemann aperte. Una presentazione completa delllargomento B ovviamente impossibile in poche pagine. Ci lin~iteremo percib solo ad enunciare il risultato ceutrale per le superficie di Riemann chiuse ($ I), risult~toche commenteremo a1 $ 2 ; quiudi a1 § 3 daremo un teoremtl di uniforn~izzazioneche B essenziale per la dimostrazione e che B di per sB interessaute. Le dimostrtlzioni saranno date per somlni capi uei $ 6, 7, 8. Una esposizione completa sarh pubblicata in seguito.

4

1.

- Enunciato del teorelna fondamentale.

I1 teorema che enunceremo assicura in sostanza la possibilith del17uniformizzazione silnultanea di tlitte le curve algebriche di dato genere g 1. Esso B dtl ritvvici~iarsia1 teore~nacorrispondente per le funzioni di Weierstrass p (#,I,z) e p' (z,l, z), 1 x 1 w, 3111 r 0 che dil l'uniformizztlzione delle cnrve di genere 1.

>

<

>

(*) Lavoro esegnito a01 oontratto No. DA - 30 - 069 - ORd - 2153 dell1#Ofice of Ordonance Research dell'Eaorcito degli Stati Uniti. Qnesto lavoro usair$ in inglese nei Rendiconti del (( Symposium on function theory r del u Tala International Institute B. I1 contennto di queato lavoro B stato esposto in nn ciolo di oonferenze tennte Eresso l a Souola Normale Superiore di Pisa dal 1a1 10 settembre 1988 in ocoasione del corso internazionale organieeato dal CIME e tennto sotto gli iluspioi della Sonola Normale Snperiore e delllIs t i t ~ ~ tM~tternatico o delllUaiversith di Pis*.

TEOREMA: Sia g > 1 un intero fissato. Esiste : 1) un dominio livzitato T nello spaxio numeric0 complesso a3g-3 (ove

z,

,...,z3g-3 sono le coordinate) omeomorfo ad tlna cells 2) ull domiuio M C (ove 2, r, , ... ,z3g-3 so110 le coordiuate) onleo-

morfo ad una oella e olomorficame~~teequivalente ad nu dominio limittlto. 3) una fuuzione continua a ( t ,r) a valori complessi, - w t m, z E T tale che a (t, z) B olomorfa in z per ogni fissato t, a (t, z) oo per z fissato e I t I m , a ( t i z) f a ( t 2 ,z) se t , t , 4 ) un gruppo 8 di antomorfisnli analitici complessi di M che opera su M senxa pnnti fissi e in modo propriamente disco~ttinuo. 5) u n groppo I' di automorfismi annlitici contplessi di T,propriaw2ente discontinuo (ma non privo di pnnti fissi). 6) nlra applicszione olomorfa z- Z(z) di P nello spnzio di Siege1 delle coppie Z = X i Y di matrici g x g simmetriche X, Y con Y > 0. ed infine 7) u n numero finito di fuazioni meromorfe FJ (2, z) definite su M automorfe rispetto a G. tali che le seguenti conclosioni siano verificate: 8) per ogui z E T le curva y (z): z = a (t,z), - w t m B 121 curve frontiera di un dominio semplicentente cow)ttnssoD (z) 11el p i a ~ ~drlla o variabile Z. 9) UII ptinto (a, z, z3g-3) = ( x , z) B in M se e solo se t E T e z E D (z). 10) ogui elellleuto di G B dell;^ forma

< <+

-

< .

,

-

+

< <+

...

ove a, b, c, 1, sono funxioui olou~ol.fein T e ad - bc = 1. 8 B generato da 29 elementi Ai A g , B, ... Bg tali c l ~ el I [ A j ,Bj] = 1 ove [ A , B] = = A B A-1 B-1 . 11) G (z), Is 6 restriziolie di G per z fissato, B 1111 gruppo di trasformazioni di Miibius del domiuio D ( z ) i n sB. La superficie di Riemauli L7 (z) = D (z)/G (z) B uua superficie clbiusa di genere g. 12) Ogni superficie di Riernauu chiusa di geuere g i! ool~formemente equivxlente ad una S(z), S(z') e S(z") sotlo conformemcute eqnivalenti se e solo se z' e r" sono equivaleuti rispetto a 13) LRmatrice Z(z) B una matrice di Rie+)tann di periodi per R(r), corrispolrde~ltead una base ilell~omologiadefinits da A i , Bi, e finirlme~~te

,... ,

, ,

r.

14) Le restrisioai delle frinzioni Fj per z fissato geuerauo il corpo delle funzioui autori~orfedi D (z) risyetto a G (z) ciob il corpo delle funxioni n~erornorfesu S (z). Osserviamo il srgnente COROLLARIO I. Lo,spazio T/I'B uuo spazio atlalitioo complesso normale. Quento Yegoe da 5) e (la 1111 teoreliit~di H. Oarttlu [9]. Diit~ostrazioui esserlzialrlre~~te diverse sollo clovute a Itiihrl [12] e Baily [4]. Baily ha ituclre dir~lostri~tocl~eqr B un aperto di Zariski di una varietil algebrica. Lo spa,zio T/I' a, com7B ovvio, lo spr~zio delle cli~ssidi superficie di Riemanu di geuere g conforriieineiib eqeivalenti (efr. -12)). COROLL ARIO 11. LO 8piczi0 M/G B uua varietil complessa., 17applicazioue ~l;ctnr'~le M / G T B olomorh, l'imnragit~eiuversa di z E T B unrb sottovitrieth regolarnietlte ininrtlrsa in M!G e coufor~iielne~~te equivalente a S(z). In modo atlalogo si possouo costrnire dei fissati comples8i su T per i qnali 1s fibra su z B S (z) x x S (z) ovvero la varietil di Jacobi di S (z).

.

...

Descrivererno in quest0 4 gli ele~tretitiuecessarii alla dimostrazioue. A) Not,azioni. La lettera, S deuoter8 IIUib superficie di Riemaun astratta. Si d i d che S ecceziouale se S animette automorfismi uon ideutici conformi ornotopi al17automorfismoideutico. Uua S non eccezionale si pub rappreseutare come U / G ove U B il setnipiano di PoiucarB e G B un grnppo Fuchsiano privo di trasformazioui ellittiche; diremo che S B di prifwa specie se i punti fissi di O souo deusi sulllasse reale. In particolare sia S di tip0 (g, n), ciob ottetiuta da uua superficie di Rie~nauuchiusa di geuere g 2 0 sopprimertdovi n 2 0 puuti (listinti. Allora 8 non B ecceziour~le(e di prima specie) se 3g - 3 n 0. Sia un differelrziale di tip0 (- 1,l)su rS, Localmeute m = p (5) dzldf ove p B uria fnazioae miearabile e 5 uua ~itliforrnizzazioue locale. PoiclrB 1 p 1 B ~ i u o soalere possinlr~o defiuire /( 9th 11 = estremo superiore essenziale di ( p 1. Se 11 1 1 ~11 1 scriveretr~o m E B (SO)e diremo che m B un d(ferenxia1e di Beltrami. In t ~ caso l ST deuote la superficie Sorn~itlitltdells strnttnw coriforme definita dalla coudizio~~e : ogui soluzione dell'equazione di Beltranui 5: = ,uf[ B tuia fuuzioue olomorfa su ST (Si richiede che la solozioue sia coutiuua ed abbia derivate geueralizzate localmeute a quadrat0 itrtegrabile). L~~rpplict~zioue uaturale S r SOsi uoterA con 1.

.

+ >

<

-

Un omeornor6smo S& So si dice quasi-cotlforme se pub essere httorizb

1

zato 11e1modo segueute h'+ S r + S O ,W esseudo conforme. Slippouiamo f qussicouforme e sia [ f j la classe d70motopia di f. Diremo ohe (S, [ f ] ,SO)B

una coppia pari. Due tali coppie (S, [f 1, So) ed (S',[f ' ] , 8;) si dirallno equih

valenti se esistouo applicazioni cor~formiS' -+ S, +So tali che [hb'f h] = = [f'] ; forte~nente equivaleu ti se Sh = $ me11tro h, p ub essere assullta eguale all'ideutittl. Oggi coppia pltri B equivalellte tld lll1a del tip0 (Sr,[l], So). Per nbuso di linguaggio iilentificlrererno spevso coppie con le corrispondenti classi di equivalenza. B) Sia So non eccezionale. L7insieme delle classi di equivale~lzeforte di coppie pari (S, [f 1, So)B lo spazio di Teichmiiller T(So). I,:I distauza di Teichmiiller tra (S,[f],No) ed (St, [gl, SO)B data (la log ((1 k)/(l - lc)) ove

,

+

k = iuf )I m 11

per

m E B (S),

(Sns,111, S) fortemeute eqnivi~le~ite a (S', [f-'g], S). Qaestn distauztl definisce una topologin su T(So). Una funzione continui~ di a valori complessi su T(So) ear& chialnnta analitica-comnplessa o oldl,aot:fa se, per ogui insieme (mi m,) c B (S), ove (8, If], So)E 2' (So),la rappresentazione di un iutorno di 0 E Cr ia C espressa d:dla (5, ,f , , )- p (f$i"'i+-.+€rn1r 7 [fI, So) @ (PI

,...,

,...

-

B olomorfa. In inodo analogo defitliremo l'analiticitd reale. U I I ~rappresentazione quasi conforme Si-$ So induce unit tazione lecita >> g* di T (S,) au 4 (So):

rappreeen-

g* dipelide soltanto da [g] e conserva la distanza di Teichmiiller e l'aualiticittl reale e complessa. I1 gruppo delle rappreser~tazio~~i lecite di T (4) in sB sltrtl denotato con r ( S O ) . Ricorrendo all'uniforlnizzazione mediante gruppi Fuchsiaui si dimostra che : T(8,) B uno spazio mnetrico conlpleto; se il gruppo fondamentale di So B generato in mod0 finito, r ( S o ) Bpropriafnente discontinuo; le funziorli aualitiche reali su T(S,,) separano i puntl. I n base 111 nuovo teorema di uniformizzazione enunciato nel $ 3 si dimostra che : ye So B di prima specie, le funzioni olomorfe su T (So) separano i punti. o) Se So B di tip0 (g, n) scriveremo T(&) = Tg,,, r ( S o )= T',, Questa notazione B giustificata dal fatto che clue qualnnque ~uperficiedi tipo (g, n) so110 quasi-conformemente equivalenti. Porremo Q = 39 - 3 n, ed assumeremo @ 0. La teoria di Teichmiiller [I, 5,13, 141 delle rappresentazioni quasi conformi estremali implica che Tg,,, sia una 2~-cella.'Iuoltre B noto che TgPnB

.

>

+

e Moduli

una varietic analitica con8plessa (cib B stato dimostrato per la prima volta da Ahlfors [a] ; cfr. anche [6, 1 0 , l l , 151). Nel nostro teorema fontl~mer~ti~le, T, I' e M tengono il I~iogo, rispettivamente, di Tg,, , Tsjoe T,,, Le osservltzior~iprecedenti giustifioaoo alouni dei nostri euiiaciati. L1esister~zadelllt rappresentazione descritt,s in (6), (13) segue, ad esempio, dalla formula variaziollale di Rauch [Ill. a) nelllenunciato del nostro teorelna,, T = Tg I I O ~ ltppare come U I I ~ variet8 a~~alitica cornplessa astmtta lntb come un dominie limitato. Questo b un caso pwrticolare di nl1 risultrcto pih generale: Tg.,,8 (olornorficamente equivalente ad) un dominie limitato in Ce. L n dimostrltzione (indicati~sommariamente in [7] b pinttosto complicata. Essa B basata sulla possibilitA di u~iiformizzareogni superficie di Rielna~iliol~iusamediuite grnppi di Schottky, ed involge unc~ltualisi geometrica dettagliata dello spazia di Schottky * di cui Tg b il ricoprimento ~u~iversale. La di~nostrltziorieprocede per induzione su g e su n ; iu tale iuduzione le superficie iperellittiche rivestono un ruolo particolare. e) la rsppresentazione di T,,, nelle forma M, ciob liella forma descritta negli enunciati 8) e 9) e lu, costruzioue del gruppo G avente le proprieth 4), lo), 11) 8 basata sul teorema di uniformiaazione del 5 3. Suppone~ldodi aver compiuto le tappe precedel~ti, IIOII 6 difficile coucludere la dirnostri~zione, ciob eostruire le fuuziorii Fj aver~tila proprieth 7), 14). Fissiamo un insierne di gelreratori Ai, Bj di G (cfr. lo)), e definiamo su ogni #(z) una base di omologia, ehe denotiamo con le stesse lettere. Sia wj il differeuzirble abeliano di prima specie avente periodo ajk su Ak (sicchb, fra I1altro, il periodo di wj su Bkb l'ele~neatoZjk di Z(z)). Sia Qjk il differenziale abeliarro di terza specie su S (z) avente periodi 0 silgli dj e tale che in ogni puuto di S(z) il residiio di Qjk eguagli I'ordiue di wj/wk. L7insieme delle funzioni [w,/wk Qjk/@1j, k, e = 1, 2, g), considerate come fnnzioni 9(z) ha le proprieta ricl~ieste. OS~ERVAZIONX. I1 teorernlt del 5 1 B sfortnuatamente di carattere piuttosto <( esiste~~ziale >>.Sarebbe utile ltvere espressioni esplicite per i domini e le funzioni ctescritte. 10 esito ad ltffermare che vi sia molta speranzlt di ottenere tali for mule^

.

,

,

...,

g 3. - Un lluovo teorelna d i nniformizzazione. Un gruppo G di trasforr~iazioliidi M6bius sarA chiltmltto quasi Fuchsiano se esiste sulla sferlt iii Riemann una curvlt di Jordan orieiitata y~ tale che y~ sia invaria~iterispetto a G, e questlultimo silt privo di puuti fissi e pro. priamente discontinuo nei domini I ( y G )e E ( y G )rispettivamente interno ed esterno a YG

.

TEOBEMAI. Siritlo 8 , e X2 due superficie di Riemanu. Supponismo che S , e 8%abbiano superficie di ricopri~neuto uuiversitle iperboliclle, e che 8, sia quasi conformemente eqaivalente i~ll'iinmagiue spec~llarej, di S2 In queste ipotesi, esiste un grappo quasi.fuchsiano G t d e c l ~ eI ( y G ) / Gsia conformemente equivalente a 8 , e E (yG)/Ga 8,. OSSI~:RVAZIONE. SB defiuita sostitue~ldociascnaa uniformizzazioue locale 5 sn 8 colt lib suil complessa coiliiigiktii $ Le ipotesi per il teorema 1 Son0 soddisfatte se S , e IE, sono chiese e dello stesso geuere 1. DIMOST~~AZIO Pol~iilluo NE. So = S z . Ne segue che El = SF per un opport,u~lom E B (So).Per ipotesi So = V/Cf0, ove Q B il semipiano soperiore e 6 , B an gruppo fucllsiauo yrivo di p ~ u ~ ~initi. ti P e r t i ~ t ~ tL/GO= o A$,,L esseutlo il sen~ipiauoinferiore. P o l ~ i a ~ lp~(z) o = 0 per 3111z < 0, e definiamo p(x)per 3115 x>O ~nedialltela cotldizione : p(x)&/dz=m. Bisults Ip(z) I ( k < l , e

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>

Esiste uuo ed nno solo omeomorfismo o, del piano in sB che lascia 0 e 1 illvariituti ed B p-oouforme, ossia B une solozione dell'equazione di Beltrami.

,

Se AE CS, 17equaziotle fuuzioni~leper iinplici~c l ~ ecop ( A (x))B uu automorfismo p-confonne dells sfera di Rieiuana, di guisa che

B uns trasforrnazione di Mobius. Si verifics olie G = w@ Go (or)-' B il groppo quasi-fuchsiaeo richiesto. Indicheremo con I la ra,ppreseiltaziolle natu1:tle di So sn go. Un omeoh morfismo S +So B detto allti-q~~asicoilfor~~ie se pub essere fattorizzato nel h modo segnente : 8 +-S:
induce uu elemento 1(@) di A, e I ( @ )= l (y) se, e soltanto se, [@I = [y], Se I (@) = 1 e se @ B anti-quasiconforme, tliremo che G rappresenta la coppia (I(yG)lG,[@I,E (yG)/G)ed ogui coppia equiville~~te iL q~~est'ultima. Riesa~rliualldola ~li~uostrazio~re del Teorelrla 1 si vede ohe di fatto abbiamo dimostrato la prima parte del TEOREVA11. Ogni coppis dispari (S, f 1, &) pub essere rappresentata da un grnppo qutlsi fiichsiarro 6, l)~lrcb& 4 ibbia I I I I ~superticie di ricoprimento universale iperbolica. Se So B di prima specie, ogni gruppo quasifuohsirt~loC, rwppr~se~lti~~lte qnesta cop~)ii~ & della forlna G, = QGQ-' Q essendo una tl~~sfor~nazione di Mobius. Per dimostri~rela secotidib p;irte dell'ellunciato possiiuno supporre che sia A!? = Sr f = 1, e d ~ ff e sia il gruppo costr~~ito diatlzi. 1,'ipotesi fatta su G, implicit l'esistenza di omeoalortismi couforn~i

,

,

con

L7ultitna relszione ilrlylica che @ = y ~ in tutti i p l ~ l ~ tflssi i di G e quindi, per colltinuittl, i l l oglri p1111todi yc; . Polliwnlo Q (x) = @(.a) per z E T(yG)UrG, Q (x) = y (x) per x E E (yc;). Q B utl automorfismo del pit~uoe G , = QCQ-1 Resta da dimostmre che Q (z) B olomorfo. Questo sarebbe iu~luediatose yc fosse rettiflcsbile. Nelle attni~licircostauze, tuttavia, dobbiamo considerare Q(z) come fuuzione di 6 = wp ( 2 ) ed utilizeitre le proprieth di w r date, ad esempio, in 131. Dw queste proprieth si trae che ntis YG = 0 e che Q(z)

.

.

a Q = 0 fuori di ha derivate generalizzate in L, iu un il~toruodi yc Poichb , at

y ~ la, analicitii di Q ovuuque segue da ben note cousiderazioni. Siamo ora in grado di costrnire il clornitrio 111 descritto nel $ 1. Neil$ dimostraziol~edel Teoretna I, sia &, I I U ~superficie prefissata, chirlsa e di genere g 1. Suppouiamo ohe per ogni

>

a5 AP 0 G abbiano lo stesao sigtriticeto di prima. Se scegliamo 8, in guisa che 0, 1, co siwuo pullti tissi, la pwrte del Teorema I1 relativa al17uuicit8 mostrs che A e G dipendono soltiinto (la r 0 uon dalla scelta particolare di m. Possialno porre G=C(r), yc =y(r), I(ya)=D(z). Vale k 11), e o(t,z)=ofi(t). I1 fatto che Ar e a dipendano olomorficamente da r segue da un risultato

provato in [3]: se u , dipende olomorficame~~te da parametri complessi, altrettanto tlccade per wr (z). Ooncludiamo enullciando due probleoii : Oglii gruppo quasi-funchsial~o poesibile dimostrsre i l Teorelntt I I( c~assiharnel~rappreseattl U I I ~coppiit ?' te $, ciob usando solbulto tribsformazioai confornii l

1. I,. V. AHLAORB, Journ. d'Ana1yae Math., 3 (194.5), pp. 1-58. 2. L. V. AHI.ROIIS,Annlytic fu~~otions, Princeton Univ. Press (1959), 45-66. 3. L. V. A H L F O Ie~ ~I.. BERN,In corso di atampn. 4. W. I,. BAILY.I n corso di stampa. 5. L. BERB, A+~alytic funolioas, Prinoeton Univ. Press, (1959), pp. 89-119. 6. L. BERS, Pmo. Int. Congr. Math. E11iobul.g 1958, pp. 309-361. 7. L. UERS, Bull. Amer. Rlath. Soc., 66 (1960), pp. 98-103. 8. L. BERS, ibiii., pp. 94-97. 9. H. CARTAN,~.efschet?z Volume, Princeton Univ. Press (1967), pp. 90-108. 10. K. KODAIRAe D. C. SPENCER,Ann. of Math., 70 (1989), pp. 14.5-166. 11. R. H. RAUCH,Proo. Nat. Aoad. 8ci. U. S. A., 41 (1955), pp. 42-49. 12. It. R ~ H R ~I .n, corso di stampa. 13. 0.TEtC1t~tft.l.E~~ Preussiscbe Akad., 22 (1940). 14. 0. TEICBMULLIER, Ibid. 4 (1943). 15. A. WELL,Sel~linnireBotlrb&ki 1958 (No. 168).